IIICh7-9.tex — March 2, 2000

P.J.M. Bongaarts Instituut Lorentz, University of Leiden bongaart@lorentz.leidenuniv.nl

Topics from 20^{th} century physics. An introductory course for students in mathematics

III. QUANTUM THEORY: CHAPTERS 7 - 9

– QUANTUM MECHANICS OF A SINGLE PARTICLE

7. Heisenberg’s uncertainty relation 7.1. Quantum mechanics of a particle: a summary. 7.2. More on position and momentum 7.3. The uncertainty relation of Heisenberg 7.4. Minimal uncertainty states 7.5. Examples 7.6. The 3-dimensional case

8. Time evolution of wave functions 8.1. Motion of a wave packet 8.2. The free particle 8.3. A particle in a box 8.4. The tunnel effect

9. The harmonic oscillator 9.1. Introduction 9.2. The classical harmonic oscillator 9.3. The quantum oscillator 9.4. Lowering and raising operators 9.5. Time evolution 9.6. Coherent states 9.7. Time evolution of coherent states

7. HEISENBERG’S UNCERTAINTY RELATION

7.1. Quantum mechanics of a particle: a summary This chapter begins the detailed discussion of the behaviour of a single particle in quantum mechanics. In this section we give a short review of basic material, some of which was discussed already in earlier sections. The Hilbert space of states H of a point particle in ordinary 3-dimensional space is L^{2}(R^{2}, dx⃗), the space of square integrable complex-valued functions on R^{3}. These functions will be denoted as ψ(x⃗), ϕ(x⃗), etc., with x⃗ = (x_{1}, x_{2}, x_{3}), and are usually called ‘wave functions’. The inner product on H is given by

(ψ_{1}, ψ_{2}) = ∫_{−∞}^{+∞} \overline{ψ_{1}(x⃗)}ψ_{2}(x⃗) dx⃗,

with dx⃗ = dx_{1}dx_{2}dx_{3}. Note that the inner product is conjugate-linear in the first variable, the standard convention in physics. The basic observable quantities of the theory are position and (linear) momentum , represented by the selfadjoint operators Q_{j} and P_{j} defined, for j = 1, 2, 3, as

(Q_{j}ψ)(x⃗) = x_{j}ψ(x⃗)

(P_{j}ψ)(x⃗) = \frac{h̄}{i} \frac{∂}{∂x_{j}} ψ(x⃗).

These operators satisfy the Heisenberg or canonical commutation relations

[P_{j}, Q_{k}] = \frac{h̄}{i}δ_{jk}

[P_{j}, P_{k}] = 0 [Q_{j}, Q_{k}] = 0,

for j, k = 1, 2, 3. Note that we consider these unbounded operators on the common dense domain S(R^{3}), the space of Schwartz functions, the C^{∞} functions that, together with all their derivatives, go to zero at infinity faster than arbitrary inverse polynomials. By this choice we avoid the more obvious domain problems.

The most important physical observable of a particle is the energy . It is represented, according to the ‘principle of canonical quantization’, by an expression in the position and momentum operators Q_{j} and P_{j} which is the same as that for the energy of a classical particle. For a particle with mass m, moving under the influence of a force F⃗ derived from a potential V as F_{j} = −\frac{∂}{∂x_{j}} V (x⃗), the most common case, this gives

Hψ = (\frac{P^{2}}{2m} + V (Q⃗)) ψ,

or, more explicitly,

(Hψ)(x⃗) = −\frac{h̄^{2}}{2m}(Δψ)(x⃗) + V (x⃗)ψ(x⃗),

with P^{2} = P_{1}^{2} + P_{2}^{2} + P_{3}^{2} and Δ the Laplace operator Δ = \frac{∂^{2}}{∂x_{1}^{2}} + \frac{∂^{2}}{∂x_{2}^{2}} + \frac{∂^{2}}{∂x_{3}^{2}} . For V = 0 the particle is called free . In this case, again the most common one, the energy operator is also the Hamiltonian , the generator of the time evolution of the system. For this reason it is denoted as H.

In classical mechanics a rotating body has angular momentum : an important physical quantity the conservation of which explains why the axis of a gyroscope persists in a given initial position and also why one is able to ride a bicycle without falling over. A classical particle moving with linear momentum p⃗ has a vector l⃗ of angular momentum with respect to the origin x⃗ = 0 of the coordinate system defined as

l⃗ = x⃗ × p⃗,

or in components

l_{1} = x_{2}p_{3} − x_{3}p_{2}
l_{2} = x_{3}p_{1} − x_{1}p_{3}
l_{3} = x_{1}p_{2} − x_{2}p_{1}.

The ‘canonical quantization principle’ gives that the quantum mechanical description three angular momentum operators L_{1}, L_{2}, L_{3}, defined as

L_{1} = Q_{2}P_{3} − Q_{3}P_{2}
L_{2} = Q_{3}P_{1} − Q_{1}P_{3}
L_{3} = Q_{1}P_{2} − Q_{2}P_{1}.

The L_{k} are differential operators, one has

L_{1} = \frac{h̄}{i} (x_{2} \frac{∂}{∂x_{3}} − x_{3} \frac{∂}{∂x_{2}} ) ,

and the two others by cyclic permutation of the indices. The three angular momentum operators do not commute. Their commutation relations can be worked out by using their explicit forms as differential operators, but can however more easily be found from the known commutation relations for the position and momentum operators. The result is

[L_{1}, L_{2}] = ih̄L_{3}
[L_{2}, L_{3}] = ih̄L_{1}
[L_{3}, L_{1}] = ih̄L_{2}

There is also the operator L^{2} = L_{1}^{2} + L_{2}^{2} + L_{3}^{2}, representing the square of the absolute value of angular momentum. It commutes with all three components L_{j}. Angular momentum in quantum mechanics has very non-classical properties: its measurement gives only discrete discrete values; its three components are incommensurable. It is also strongly connected with symmetry aspects: the three operators L_{j} span a representation of the Lie algebra of SO(3), the 3-dimensional rotation group. All this will be discussed later.

7.2. More on position and momentum When discussing the properties of position, momentum and energy in quantum mechanics, it is an inessential but notationally very convenient simplification to do this first for a 1-dimensional particle, a simplification which would of course not make sense for the discussion of angular momentum. For a 1-dimensional particle the state space H is the Hilbert space L^{2}(R^{1}, dx), with inner product

(ψ_{1}, ψ_{2}) = ∫_{−∞}^{+∞} \overline{ψ_{1}(x)}ψ_{2}(x) dx.

As basic observables one has the position and momentum operators Q and P, defined as

(Qψ)(x) = xψ(x)

and

(P ψ)(x) = \frac{h̄}{i} \frac{d}{dx}ψ(x),

satisfying the Heisenberg commutation relation

[P, Q] = \frac{h̄}{i} 1.

The energy operator is

H = \frac{P^{2}}{2m} = −\frac{h̄^{2}}{2m} \frac{d^{2}}{dx^{2}} ,

for a free particle, and

H = \frac{P^{2}}{2m} + V (Q) = −\frac{h̄^{2}}{2m} \frac{d^{2}}{dx^{2}} + V (x),

for a particle moving in a potential V. We take the space of Schwartz functions S(R^{1}) as common dense domain on which the P and Q are essentially selfadjoint. The operator H is also essentially selfadjoint on this domain if the potential V is a bounded real function on R^{1}; it is a nontrivial mathematical problem to decide for which unbounded functions V this remains true.

A useful device is the Fourier transform

ψ̂(p) = \frac{1}{(2πh̄)^{1/2}} ∫_{−∞}^{+∞} ψ(x)e^{− \frac{i}{h̄} px} dx,

with the inverse transform

ψ(x) = \frac{1}{(2πh̄)^{1/2}} ∫_{−∞}^{+∞} ψ̂(p)e^{+ \frac{i}{h̄} px} dp.

For the ψ and its Fourier transform ψ̂ one has Plancherel’s formula

∫_{−∞}^{+∞} |ψ(x)|^{2} dx = ∫_{−∞}^{+∞} |ψ̂(p)|^{2} dp.

The theory of Fourier transforms of functions of a single variable is a fairly subtle part of mathematical analysis. In this case technicalities can be avoided: as a linear transformation of the Hilbert space the Fourier transform makes sense as an integral which defines an invertible continuous linear map from S(R) onto itself and extends to a unitary operator on the full Hilbert space of all square integrable functions.

In our general outline of the formalism of quantum theory and its physical interpretation we mentioned already that the square of the absolute value of the wave function ψ(x), provided that it is normalized to one, i.e. provided that its Hilbert space norm is one, is interpreted as the probability density for the position of the particle: the probability for finding the particle in an interval [x_{1}, x_{2}] is given by the integral ∫_{x_{1}}^{x_{2}} |ψ(x)|^{2}dx. The mean value, the expectation, or expectation value as one says in the physics literature, is given by (\overline{Q})_{ψ} = ∫_{−∞}^{+∞} x|ψ(x)|^{2}dx, and, more generally, the n^{th} moment as (\overline{Q^{n}})_{ψ} = ∫_{−∞}^{+∞} x^{n}|ψ(x)|^{2}dx, in agreement with the general Hilbert space formula (\overline{A^{n}})_{ψ} = (ψ, A^{n}ψ), for the measurement of an observable A in a state ψ.

The same operator formula holds for the expectation value and the n^{th} moments of the probability distribution for momentum measurements in the state ψ(x): (\overline{P^{n}})_{ψ} = (ψ, P ψ). We use this to calculate (\overline{P^{n}})_{ψ} explicitly. We note first that Plancherel’s formula can be generalized to

∫_{−∞}^{+∞} ψ_{1}^{∗}(x)ψ_{2}(x)dx = ∫_{−∞}^{+∞} ψ̂_{1}^{∗}(p)ψ̂_{2}(p)dp,

for arbitrary functions ψ_{1} and ψ_{2} in L^{2}(R^{1}, dx), and with ψ̂_{1} and ψ̂_{2} the Fourier transforms of ψ_{1} and ψ_{2}. To prove this one uses a general expression for the inner product (ψ_{1}, ψ_{2}) in terms of (ψ_{1} ± ψ_{2}, ψ_{1} ± ψ_{2}) and (ψ_{1} ± iψ_{2}, ψ_{1} ± iψ_{2}).

We write, for a function ψ(x) in S(R),

(P ψ)(x) = \frac{h̄}{i} \frac{d}{dx} [\frac{1}{(2πh̄)^{1/2}} ∫_{−∞}^{+∞} ψ̂(p)e^{\frac{i}{h̄} px}dp] =

= \frac{1}{(2πh̄)^{1/2}} ∫_{−∞}^{+∞} pψ̂(p)e^{\frac{i}{h̄} px}dp,

which implies (P̂ ψ)(p) = pψ̂(p), and more generally (P̂^{n}ψ)(p) = p^{n}ψ̂(p). Using the generalization of Plancherel’s formula gives

(\overline{P^{n}})_{ψ} = (ψ, P^{n}ψ) = ∫_{−∞}^{+∞}(P^{n}ψ)^{∗}(x)ψ(x)dx =
= ∫_{−∞}^{+∞}(P̂^{n}ψ)^{∗}(p)ψ̂(p)dp = ∫_{−∞}^{+∞} p^{n}|ψ̂(p)|^{2}dp,

which means that the function |ψ̂(p)|^{2} is the probability density for the measurement of momentum p.

The function ψ_{p}(x) = e^{\frac{i}{h̄} px} is a plane wave in (one-dimensional) space, with wave length λ = 2πh̄/p. It is an eigenfunction of the momentum operator P = \frac{h̄}{i} \frac{∂}{∂x} for the eigenvalue p. Physicists therefore use such plane waves for the quantum mechanical description of a particle with sharp momentum p. This expresses one of the oldest and most influential general ideas of quantum mechanics, the particle-wave duality : a point particle of mass m and momentum p = mv is described by a plane wave with wave lenght

λ = \frac{2πh̄}{p}.

This λ is called the De Broglie wave length of the particle, after the French physicist Louis de Broglie, one of the pioneers of early quantum mechanics, who was the first to suggest this idea of a ‘particle - wave duality’, as was already mentioned in Section 2.1. of Chapter 2. The interpretation of the function ψ_{p}(x) = e^{\frac{i}{h̄} px} as a physical state is nevertheless incorrect. The general formalism of quantum theory tells us that a function ψ(x), which is an eigenfunction of a selfadjoint operator A for an eigenvalue a, should be interpreted a physical state with a as a sharp value for the observable represented by A. However, the function ψ(x) should be square integrable. The plane wave ψ_{p}(x) = e^{\frac{i}{h̄} px} is clearly not square integrable, it is not an element of the Hilbert space of physical states.

The Fourier transformation provides us with a mathematically more precise way of expressing the above physical idea. The intuitive meaning of the Fourier formula for the function ψ(x) – the ‘wave function’ as it is called, not surprisingly in view of what has just been said – is that it gives ψ(x) as a ‘continuous superposition’ of plane waves. To approximate the notion of a sharp momentum state we can use a superposition over a narrow p-interval ε around a p-value p_{0}, for instance

ψ_{p_{0},ε}(x) = ε^{− \frac{1}{2}} ∫_{p_{0}− \frac{1}{2} ε}^{p_{0}+ \frac{1}{2} ε} e^{\frac{i}{h̄} px}dp.

The constant ε^{\frac{1}{2}} in front is such that ψ_{p_{0},ε} will be normalized to one. An elementary calculation gives

ψ_{p_{0},ε}(x) = F_{ε}(x) e^{\frac{i}{h̄} p_{0}x},

with

F_{ε}(x) = ε^{\frac{1}{2}} \frac{sin \frac{ε}{2h̄}x}{\frac{ε}{2h̄}x}.

The function ψ_{p_{0},ε}(x) is a plane wave ψ_{p_{0}}(x), multiplied with the real function F_{ε}(x) which makes it square integrable. F (x) has a maximum ε^{\frac{1}{2}} in x = 0 and goes to 0 for x going to ±∞, oscillating with decreasing amplitude between the absolute value of the function \frac{1}{x}. For small ε, i.e. when the momentum lies within a narrow interval, the function flattens out in space, which means physically that the position of the particle becomes very unpredictable. For large ε, i.e. for undeterminate momentum, F_{ε}(x) becomes sharply peaked around x = 0, which means that the position of the particle becomes narrowly determined. This example shows us that we can have quantum mechanical states in which the position of the particle can be predicted with arbitrary precision, that we can also have states in which the momentum is as sharply defined as one wishes, but that we can not have both at the same time. This can be shown generally by means of an inequality for the standard deviations of the measurements of position and momentum which will be discussed in the next section.

The fact that one cannot simultaneously measure the position and velocity of a particle with arbitrary precision means that the particle can be observed at discrete times, but not over a continuous time interval. It cannot be followed in time; its orbit is not observable. There is an even stronger interpretation of this: the classical notion of orbit of particle is meaningless in quantum mechanics .

7.3. The uncertainty relation of Heisenberg Consider for a moment an arbitrary quantum system with Hilbert space H. Take two physical observables a and b, not necessarily commensurable, represented by two (not necessarily commuting) selfadjoint operators A and B. Let ψ be a state vector in H, which should be such that Aψ, A^{2}ψ, Bψ, B^{2}ψ, ABψ and BAψ are well-defined. Denote in the usual way the mean values ⟨A⟩_{ψ} = (ψ, Aψ) and ⟨B⟩_{ψ} = (ψ, Bψ) as \overline{a} and \overline{b}, and the standard deviations ⟨(A − \overline{a})^{2}⟩^{1/2} = (ψ, (A − \overline{a})^{2}ψ)^{1/2} and ⟨(B − \overline{b})^{2}⟩^{1/2} = (ψ, (B − \overline{b})^{2}ψ)^{1/2} as Δa and Δb. We then have

|(ψ, [A, B]ψ)| = |(ψ, [A − \overline{a}, B − \overline{b}]ψ)| =
= 2 |Im ((A − \overline{a})ψ, (B − \overline{b})ψ) | ≤ 2 | ((A − \overline{a})ψ, (B − \overline{b})ψ) |.

The inequality of Schwarz gives

2 | ((A − \overline{a})ψ, (B − \overline{b})ψ) | ≤ 2 ||(A − \overline{a})ψ|| ||(B − \overline{b})ψ|| =
= 2 (ψ, (A − \overline{a})^{2}ψ)^{1/2} (ψ, (B − \overline{b})^{2}ψ)^{1/2} = 2 Δa Δb.

The result is the general uncertainty relation , for the standard deviations of the observables a and b in the state ψ,

(Δa)_{ψ} (Δb)_{ψ} ≥ \frac{1}{2} |(ψ, [A, B]ψ)|.

This is a restriction on the possibility of simultaneously measuring a and b. For commensurable a and b, when A and B commute, there is of course no restriction.

Return now to the case of a 1-dimensional particle and choose A = P and B = Q. This gives immediately the Heisenberg uncertainty relation

Δp Δx ≥ \frac{h̄}{2}.

Note that we write Δx and Δp instead of Δx_{ψ} and Δp_{ψ} because the righthand side no longer depends on the state vector ψ. The Heisenberg uncertainty relation is one of the centre pieces of elementary quantum mechanics, illustrating the fundamental incommensurability of position and momentum measurements.

7.4. Minimal uncertainty states For an arbitrary state ψ the product of the standard deviations in position and momentum is always larger or equal than \frac{1}{2}h̄. We are interested in states ψ for which this product is minimal , i.e. equal to \frac{1}{2}h̄. Such states of minimal uncertainty in position and momentum will just be called minimal states or minimal wave packets in this section. Finding all minimal states is simplified by the following remarks: a. Translating a wave function ψ over a distance a in space gives a new wave function ψ_{a}(x) = ψ(x − a). Its probability density for measuring position is ρ_{a}(x) = |ψ_{a}(x)|^{2} = |ψ(x − a)|^{2} = ρ(x − a), which gives a translated expectation value (\overline{x})_{a} = \overline{x} + a. Its standard deviation Δx clearly remains the same. The Fourier expression for the translated function ψ_{a} is

ψ_{a}(x) = ψ(x − a) = \frac{1}{(2π)^{1/2}} ∫_{−∞}^{+∞} ψ̂(p)e^{\frac{i}{h̄} p(x−a)}dp

= \frac{1}{(2π)^{1/2}} ∫_{−∞}^{+∞} ψ̂_{a}(p)e^{\frac{i}{h̄} px}dp,

with the new Fourier transform

ψ̂_{a}(p) = e^{− \frac{i}{h̄} pa}ψ̂(p).

This is just multiplication of ψ̂(p) with a phase factor, which does not affect ρ̂(p) = |ψ̂(p)|^{2}, the probability density for the measurement of moment, and consequently leaves both the expectation value \overline{p} and the standard deviation Δp unchanged. This means that translation in x carries minimal states into minimal states. Note that the above Fourier formula argument shows that translation in space of a wave function ψ over a distance a can be seen as the result of the action of the 1-parameter unitary group U(a) = e^{− \frac{i}{h̄} aP}, or

ψ_{a} = e^{− \frac{i}{h̄} aP} ψ.

b. A similar argument applies to a translation in momentum space, given in terms of the Fourier transform as ψ̂_{b}(p) = ψ̂(p − b), for an arbitrary b in R^{1}. The mean value \overline{p} changes into \overline{p} + b, but the standard deviation Δp remains the same. The wave function in x-space is multiplied by a phase factor,

ψ_{b}(x) = e^{\frac{i}{h̄} bx}ψ(x),

so it is clear that neither \overline{x} nor Δx changes. This means that translation in p also maps minimal states into minimal states. Note that these translations are the result of the action of a second 1-parameter group of unitary operators, n.l. the group V (b) = e^{\frac{i}{h̄} bQ}, or

ψ_{b} = e^{\frac{i}{h̄} bQ}ψ.

The remarks a. and b. together imply that it is sufficient to find the minimal states with \overline{x} = 0 and \overline{p} = 0; all others can be obtained from these by the action of the operators U(a) = e^{− \frac{i}{h̄} aP} and V (b) = e^{\frac{i}{h̄} bQ}, for all possible real a and b.

The derivation of Heisenberg’s uncertainty relation in Section 7.3 gives for a state ψ with Δp = Δx = 0 the relations

h̄ = |(ψ, [P, Q]ψ)| ≤ 2|(P ψ, Qψ)| ≤ 2||P ψ|| ||Qψ|| = 2Δp Δx.

For a minimal state, i.e. with Δp Δx = \frac{1}{2}h̄, the Schwarz inequality in the middle becomes an equality. This is only possible if P ψ = λQψ, for some complex number λ. One has on one hand ||P ψ|| = |λ| ||Oψ||, or |λ| = \frac{Δp}{Δx}, and on the other hand Δp Δx = \frac{1}{2}h̄ = \frac{i}{2}(ψ, [P, Q]ψ) = \frac{i}{2}Im (P ψ, Qψ) = (Δx)^{2}Im λ, which gives Im λ = \frac{Δp}{Δx}. Together this gives λ = i\frac{Δp}{Δx}. The minimal states with Δp = Δx = 0 are therefore normalized vectors ψ satisfying

P ψ = i \frac{Δp}{Δx} Qψ,

i.e. solutions of the differential equation

(\frac{d}{dx} + \frac{1}{h̄} \frac{Δp}{Δx} x) ψ(x) = 0.

The solutions of this equation are obtained by the substitution y = x^{1/2}, which leads to an even more elementary differential equation in y. One finds as general solution

ψ(x) = C e^{− \frac{ϱ}{2h̄} x^{2}},

with the positive parameter ϱ = \frac{Δp}{Δx}. The constant C can be determined, up to an irrelevant phase factor, by the normalization condition (ψ, ψ) = 1. Using the well-known Gaussian integral ∫_{−∞}^{+∞} e^{−t^{2}}dt = \sqrt{π} one finds C = (\frac{ϱ}{πh̄})^{1/4}. With this we have a 1-parameter set of minimal states with Δp = Δp = 0. Their wave functions are

ψ_{ϱ}(x) = (\frac{ϱ}{πh̄})^{1/4} e^{− \frac{ϱ}{2h̄} x^{2}},

with the positive parameter ϱ equal to the quotient \frac{Δp}{Δx}. The probability density for the measurement of position is

|ψ_{ϱ}(x)|^{2} = (\frac{ϱ}{πh̄})^{\frac{1}{2}} e^{− \frac{ϱ}{h̄} x^{2}} = \frac{1}{(2π)^{1/2}Δx} e^{− \frac{1}{2(Δx)^{2}} x^{2}},

a normal or Gaussian distribution centered at x = 0 and with standard deviation Δx. One can show that the Fourier transform of ψ_{ϱ} is

ψ̂_{ϱ}(p) = (\frac{1}{πh̄ϱ})^{1/4} e^{− \frac{1}{2ϱh̄} x^{2}},

which leads to a probability density for the measurement of momentum

|ψ̂_{ϱ}(p)|^{2} = (\frac{1}{πh̄ϱ})^{1/2} e^{− \frac{1}{ϱh̄} p^{2}} = \frac{1}{(2π)^{1/2}Δp} e^{− \frac{1}{2(Δp)^{2}} p^{2}},

a Gaussian distribution centered at p = 0 and with standard deviation Δp.

The minimal states with arbitrary values for \overline{p} and \overline{x} are obtained from those with \overline{p} = \overline{x} = 0 by translations in position and momentum space. The result is a 3-parameter collection of wave functions

ψ_{\overline{p},\overline{x},ϱ}(x) = (\frac{ϱ}{πh̄})^{1/4} e^{\frac{i}{h̄} \overline{p}(x−\overline{x})}e^{− \frac{ϱ}{2h̄} (x−\overline{x})^{2}},

with Fourier transform

ψ̂_{\overline{p},\overline{x},ϱ}(p) = (\frac{1}{πh̄ϱ})^{1/4} e^{− \frac{i}{h̄} \overline{x}p} e^{− \frac{1}{2h̄ϱ} (p−\overline{p})^{2}}.

The probability densities for position and momentum measurements are

|ψ_{\overline{p},\overline{x},ϱ}(x)|^{2} = (\frac{ϱ}{πh̄})^{1/2} e^{− \frac{1}{ϱh̄} (x−\overline{x})^{2}} = \frac{1}{(2π)^{1/2}Δx} e^{− \frac{1}{2(Δx)^{2}} (x−\overline{x})^{2}}

|ψ̂_{\overline{p}\overline{x},ϱ}(p)|^{2} = (\frac{1}{πh̄ϱ})^{1/2} e^{− \frac{1}{h̄ϱ} (p−\overline{p})^{2}} = \frac{1}{(2π)^{1/2}Δp} e^{− \frac{1}{2(Δp)^{2}} (p−\overline{p})^{2}},

again Gaussian distributions with standard deviations Δx and Δp, now centered at x = \overline{x} and p = \overline{p}.

7.5. Examples The numerical consequences of the uncertainty relation can be demonstrated in appropiately chosen concrete physical systems.

a. An example of a microscopic system: The mass of an electron is roughly 9 × 10^{−31} kg. Assume that it moves with a velocity v = 10^{3} m/s, so the momentum p is 9 × 10^{−28} kg.m/s. (We use KMS (kilogram - meter - second) units.) Suppose again an uncertainty in the momentum of 0.1 percent, so Δp = 9 × 10^{−31} kg.m/s. This gives for the uncertainty in the position a value Δx = 4.5 × 10^{−5} m. This may be a small lenght in the macroscopic world, but it is very large in the atomic contex, for instance in comparison with the size of the hydrogen atom, in a certain way defined by the radius of the inner electron orbit which is roughly 5 × 10^{−11} m. (With this ‘orbit’ we mean the space region in which the wave function of the electron in its state of lowest energy is appreciably different from 0. An orbit in the classical sense does not exist, as was discussed in 7.2.)

b. An example of a macroscopic system:

Think of a bullet with a size of a centimeter and mass of 10 grammes, moving with a velocity of 200 meters per second. (The velocity of sound in air at room temperature is 340 meters per second). The momentum p = mv is 2 kg.m/s. Suppose that there a is an uncertainty in this momentum of 0, 1 percent, i.e. the standard deviation Δp is equal to 2 × 10^{−3} kg.m/s. Planck’s constant h̄ is 1.054 × 10^{−34} kg.m^{2}/s, so \frac{1}{2}h̄ is roughly 5 × 10^{−35} kg.m^{2}/s. This gives the minimal uncertainty in the position of the bullet as Δx = \frac{1}{2}h̄/Δp = 2.5 × 10^{−33} m, a value which is extremely small and would be totally unobservable in this context.

The last example illustrates an important general point in the application of quantum theory to physics. Quantum theory is considered to be valid for all physical phenomena. It is indispensable for understanding the world of atomic physics. However, in macroscopic physical situations its predictions can often not be distinguished experimentally from those of classical mechanics. In such cases the continued use of classical ideas is justified.

7.6. The 3-dimensional case The basic formulas for the quantum mechanical description of a 3-dimensional particle were given in 7.1. In the subsequent sections we discussed the case of a particle in 1-dimensional space. This simplified our formulas, while nothing essential was lost. In this section we briefly review some of the remaining aspects of the 3-dimensional case and give – for the sake of completenesss – the 3dimensional versions of the appropriate formulas.

The position operators Q_{1}, Q_{2}, Q_{3} commute; they form a triple of commensurable operators , which means that they give a system of stochastic variables in the sense of ordinary probability theory: a normalized state vector ψ defines the joint probability density

ρ(x_{1}, x_{2}, x_{3}) = |ψ(x_{1}, x_{2}, x_{3})|^{2}

for the three position variables x_{1}, x_{2}, x_{3}. The same is true for the three components of momentum, with a simultaneous probability density

ρ̂(p_{1}, p_{2}, p_{3}) = |ψ̂(p_{1}, p_{2}, p_{3})|^{2}.

For this one needs the 3-dimensional Fourier transformation. The formulas for this transformation and its inverse are

ψ̂(p⃗) = \frac{1}{(2πh̄)^{3/2}} ∫_{−∞}^{+∞} ψ(x⃗) e^{− \frac{i}{h̄} p⃗·x⃗}dp⃗

ψ̂(x⃗) = \frac{1}{(2πh̄)^{3/2}} ∫_{−∞}^{+∞} ψ̂(x⃗) e^{\frac{i}{h̄} p⃗·x⃗}dx⃗,

with the Plancherel formula

∫_{−∞}^{+∞} |ψ(x⃗|^{2}dx⃗ = ∫_{−∞}^{+∞} |ψ̂(p⃗|^{2}dp⃗.

Each pair (P_{j}, Q_{j}), j = 1, 2, 3, consists of two incommensurable observables. There are no simultaneous probability distributions; instead we have, for j = 1, 2, 3, the Heisenberg uncertainty relations

Δp_{j} Δx_{j} ≥ \frac{1}{2}h̄.

The set of states of minimal uncertainty in position and momentum, states which satisfy

Δp_{j} Δx_{j} = \frac{1}{2}h̄,

for j = 1, 2, 3, consists of all product wave functions

ψ_{\overline{p_{1}},\overline{x_{1}},ϱ_{1}}(x_{1}) ψ_{\overline{p_{2}},\overline{x_{2}},ϱ_{2}}(x_{2}) ψ_{\overline{p_{3}},\overline{x_{3}},ϱ_{3}}(x_{3}),

for abritrary expectation values \overline{p_{j}}, \overline{x_{j}} and arbitrary positive values of the quotients ϱ_{j} = Δp_{j}/Δx_{j}, for j = 1, 2, 3.

8. TIME EVOLUTION OF WAVE FUNCTIONS

8.1. Motion of a wave packet In the preceding chapter we discussed the quantum mechanical description of a free particle at a fixed time. In this section its time evolution will be studied. For simplicity we again consider first the case of a particle in 1-dimensional space. In general the time evolution of a state in quantum theory, from an initial time t_{1} to a later time t_{2}, is given by a 1-parameter group of unitary operators U(t) = e^{− \frac{i}{h̄} tH} as

ψ(t_{2}) = e^{− \frac{i}{h̄} (t_{2}−t_{1})H}ψ(t_{1}).

This means that state vectors, as functions of time, should be solutions of the ‘abstract’ Schrödinger equation

\frac{d}{dt}ψ(t) = −\frac{i}{h̄}Hψ(t),

which in the special case of a 1-dimensional particle becomes the ‘concrete’ Schrödinger equation

\frac{∂}{∂t}ψ(x, t) = −\frac{i}{h̄} (−\frac{h̄^{2}}{2m} \frac{∂^{2}}{∂x^{2}} + V (x)) ψ(x, t),

with the state vector now the ‘wave function’ ψ(x, t). This differential equation is of course only defined for wave functions in the domain of the (unbounded) operator H, for instance for functions that are in S(R) as functions of x.

A state of a particle may have a wave function which is different from 0 only in a small region of space and which has a Fourier transform concentrated in a small momentum interval. It will have small standard deviations Δx and Δp from the average values \overline{x} and \overline{p} of position and momentum. Such a state can be called ‘quasi classical’. It resembles a classical particle with well-defined position x = \overline{x} and momentum p = \overline{p}. This is of course only approximately so, because Heisenberg’s uncertainty relation tells us that Δx and Δp cannot be both arbitrary small; the product Δp Δx has \frac{1}{2}h̄ as lower limit. The minimal uncertainty states discussed in Section 7.4 are the best one can have in this respect.

In fact, even the time evolution of an arbitrary wave function has classical aspects. This is in particular true for the time dependence of the average values of position and momentum. One has for the expectation value of position

\frac{d\overline{x}(t)}{dt} = \frac{d}{dt}(ψ(t), Qψ(t)) =

= (−\frac{i}{h̄}Hψ(t), Qψ(t)) + (ψ(t), −\frac{i}{h̄}QHψ(t)) =

= \frac{i}{h̄}(ψ(t), [H, Q]ψ(t)).

Using the basic commutation relation [P, Q] = \frac{i}{h̄}1, together with the fact that Q commutes with V (Q), one finds

[H, Q] = [\frac{P^{2}}{2m} + V (Q), Q] = \frac{1}{2m}[P^{2}, Q] = \frac{h̄}{im}P,

which gives finally

\frac{d\overline{x}(t)}{dt} = \frac{1}{m}(ψ(t), P ψ(t)) = \frac{1}{m}\overline{p}(t).

For \overline{p}(t) one obtains in a similar manner

\frac{d\overline{p}(t)}{dt} = \frac{i}{h̄}(ψ(t), [H, P]ψ(t)) = −(ψ(t), \frac{dV}{dQ}ψ(t)) = −\overline{[\frac{dV}{dQ}]},

in which \frac{dV}{dQ} is the operator which multiplies a wave function with the function

\frac{dV (x)}{dx}. The pair of first order equations

m\frac{d\overline{x}}{dt} = \overline{p}, \frac{d\overline{p}}{dt} = −\overline{[\frac{dV}{dx} ]}

can be combined to single second order equation

m\frac{d^{2}\overline{x}}{dt^{2}} = −\overline{[\frac{dV}{dx} ]},

which brings out even more clearly the analogy with the classical equations of motion. The time evolution of a wave function which is quasi-classical at an initial time will resemble in first approximation the motion of a classical particle with position x = \overline{x}(t) and momentum p = \overline{p}(t), at least for a certain initial period of time, because the wave packet will eventually spread out, and its quasi-classical character will then be lost.

8.2. The free particle In this simple case the Schrödinger equation can be explicitly solved, i.e. the wave function at time t_{2} can be expressed in the wave function at an earlier time t_{1}. For this one can use the Fourier transformation. We demonstrate this for the 1-dimensional case. Let the initial state at t = t_{1} be given by the function ψ(x, t_{1}), a function in which we assume to be in S(R^{1}) in the variable x. Its Fourier transform is

ψ̂(p, t_{1}) = \frac{1}{(2πh̄)^{1/2}} ∫_{−∞}^{+∞} ψ(x, t_{1}) e^{− \frac{i}{h̄} px}dx.

The Hamiltonian is a simple expression in the momentum operator P, so the action of the unitary evolution operator e^{− \frac{i}{h̄} tH} in terms of the Fourier transform is also simple and gives

ψ̂(p, t_{2}) = e^{− \frac{i}{h̄} (t_{2}−t_{1}) \frac{p^{2}}{2m}} ψ̂(p, t_{1}).

By using the inverse Fourier transformation finally one obtains the solution at t = t_{2} as

ψ(x, t_{2}) = \frac{1}{(2πh̄)^{1/2}} ∫_{−∞}^{+∞} ψ̂(p, t_{2}) e^{\frac{i}{h̄} px}dp =

= \frac{1}{(2πh̄)^{1/2}} ∫_{−∞}^{+∞} ψ̂(p, t_{1}) e^{− \frac{i}{h̄} (t_{2}−t_{1}) \frac{p^{2}}{2m}} e^{\frac{i}{h̄} px}dp =

= \frac{1}{2πh̄} ∫_{−∞}^{+∞} dp ∫_{∞}^{+∞} dx′ ψ(x′, t_{1}) [e^{− \frac{i}{h̄} p(x−x′)} e^{− \frac{i}{h̄} (t_{2}−t_{1}) \frac{p^{2}}{2m}} ] .

This is a well-defined repeated integral which gives ψ(x, t_{2}) as a function in S(R^{1}) for all t_{2} ≥ t_{1}. It is sometimes written as

ψ(x, t_{2}) = ∫_{−∞}^{+∞} G(x, t_{2}; x′, t_{1})ψ(x′, t_{1})dx′,

with

G(x, t_{2}; x′, t_{1}) = \frac{1}{2πh̄} ∫_{−∞}^{+∞} e^{− \frac{i}{h̄} p(x′−x)} e^{− \frac{i}{h̄} (t_{2}−t_{1}) \frac{p^{2}}{2m}} dp,

the Green’s function or evolution kernel of the differential equation. This involves an interchange in the order of integrations which is strictly speaking not allowed and leads to an integral for G which is singular; the formula, which is very useful, can however be made rigorous by interpreting G as a distribution or generalized function in the sense of Laurent Schwartz, instead of a as function in the ordinary sense.

The analogous formula for the 3-dimensional case can be written down immediately as

ψ(x⃗, t_{2}) = \frac{1}{(2πh̄)^{3}} ∫_{−∞}^{+∞} dp⃗ ∫_{−∞}^{+∞} dx′ ψ(x′, t_{1}) [e^{− \frac{i}{h̄} p⃗·(x⃗−x′)} e^{− \frac{i}{h̄} (t_{2}−t_{1}) \frac{p^{2}}{2m}} ] ,

with the heuristic reformulation

ψ(x⃗, t_{2}) = ∫_{−∞}^{+∞} G(x⃗, t_{2}; x′, t_{1})ψ(x′, t_{1})dx′,

with

G(x⃗, t_{2}; x′, t_{1}) = \frac{1}{(2πh̄)3} ∫_{−∞}^{+∞} e^{− \frac{i}{h̄} p⃗·(x′−x⃗)} e^{− \frac{i}{h̄} (t_{2}−t_{1}) \frac{p^{2}}{2m}} dp⃗.

Note that the equations for the time evolution of average position and momentum are now precisely the classical equation for position and momentum, because V (x) = 0.

8.3. A particle in a box Consider the quantum mechanical description of a 1-dimensional particle moving freely in a box of length a. The Hilbert space is H = L^{2}([0, a], dx). The position operator Q : ψ(x) ↦ xψ(x) is now defined on all vectors in H; it is a bounded operator with norm ||Q|| = a. The momentum operator P, the differential operator \frac{h̄}{i} \frac{d}{dx}, is well-defined on the linear subspace D^{(1)} of all differentiable functions ψ in L^{2}([0, a], dx), with the derivative ψ^{(1)}(x) = \frac{d}{dx}ψ(x) again in L^{2}([0, a], dx), but it is clearly not symmetric on D^{(1)}, because the condition for symmetry on a domain, (ψ_{1}, P ψ_{2}) = (P ψ_{1}, ψ_{2}), for all pairs of vectors ψ_{1}, ψ_{2} from that domain, implies the boundary condition

\overline{ψ_{1}(a)}ψ_{2}(a) = \overline{ψ_{1}(0)}ψ_{2}(0).

This leads to the possible domains of symmetry

D_{α}^{(1)} : = {ψ ∈ D^{(1)} | ψ(a) = e^{iα}ψ(0)},

for each α ∈ [0, 2π). The restriction of P to each D_{α}^{(1)} gives a selfadjoint operator P_{α}. One checks that is has a purely discrete spectrum, with eigenvalues

p_{n}^{α} = \frac{h̄}{a}(α + 2π),

and a corresponding orthonormal basis of eigenfunctions

ϕ_{n}^{α}(x) = \frac{1}{\sqrt{a}}e^{\frac{i}{h̄} p_{n}^{α}x},

for n = 0, ±1, ±2, . . .. Note that P defined on the domain consisting of all functions ψ which vanish in x = 0 and x = a would be too small. The restriction of P to this domain would be symmetric but not selfadjoint. This gives us a simple example of a symmetric operator which is not selfadjoint, but has (an infinite number of) selfadjoint extensions.

For the discussion of the physical interpretation of this situation we consider the possibilities for the energy operator, restrictions of the differential operator H = −\frac{h̄^{2}}{2m} \frac{d^{2}}{dx^{2}} to suitable domains. Starting from the P^{α} we find selfadjoint operators H_{α} = \frac{(P_{α})^{2}}{2m} defined on the domains D_{α}^{(2)} consisting of the functions ψ in L^{2}([0, a], dx) that have second derivatives which are again in L^{2}([0, a], dx) and satisfy moreover the boundary conditions ψ(a) = e^{iα}ψ(0) and ψ^{(1)}(a) = e^{iα}ψ^{(1)}(0). They have the same orthonormal systems {ϕ_{n}^{α}}_{n} of eigenvectors as the P_{α}. The energy eigenvalues are

E_{n}^{α} = \frac{(p_{n}^{α})^{2}}{2m} = \frac{h̄^{2}}{2ma^{2}} (α + 2πn)^{2},

for n = 0, ±1, ±2, . . .. There is an additional selfadjoint energy operator H^{0}, n.l. the restriction of H = −\frac{h̄^{2}}{2m} \frac{d^{2}}{dx^{2}} to the space D^{(2),0} of all ψ in L^{2}([0, a], dx), having second derivative ψ^{(2)} in L^{2}([0, a], dx), and with boundary condition ψ(a) = ψ(0) = 0. H^{0} has an orthonormal basis of eigenfunctions

ψ_{n}^{0}(x) = \sqrt{\frac{2}{a}} sin (\frac{nπx}{a}) ,

with eigenvalues

E_{n}^{0} = \frac{h̄^{2}}{2m} (\frac{πn}{a})^{2} ,

for n = 1, 2, . . ..

Taking H_{α} for α = 0 as energy operator means that when the particle reaches one endpoint of the interval [0, a] it immediately reappears at the other endpoint. The motion is periodic and the points x = 0 and x = a can be identified; the particle moves on a circle. For α ≠ 0 the particle passes a barrier at x = 0 = a, at which it picks up a phase, as is usual in the transmission of waves. The case with energy operator H^{0} describes a particle moving in box in the ordinary sense; it is reflected at the walls of the box and the sign of its velocity changes instantaneously. Note that in this case no proper selfadjoint momentum operator can be defined. This is because we have in a certain sense a singular limit, even in the classical case, with an infinite force at the wall driving the particle back, a limit of a finite repulsive force over a short interval near the wall. In any case, this quantum mechanical model of a particle in a box illustrates in a simple way the physical meaning of different selfadjoint extensions of a symmetric operator.

8.4. The tunnel effect Consider a 1-dimensional particle moving in a potential V (x) which vanishes for all real x, except in two intervals, [−(a + d), −a] and [a, a + d], on which one has V (x) = E_{0}, with E_{0}, a and d positive constants and d much smaller than a, i.e. a potential V (x) which consists of two walls of hight E_{0} and thickness d. In the classical description a particle with kinetic energy \frac{p^{2}}{2m} < E_{0} cannot pass the potential walls. If it is inside the interval [−a, a] it will remain inside, when it is outside it will remain outside. A particle with kinetic energy larger than E_{0} can move freely along the real line; it passes over the walls, its velocity only momentarily perturbed by the potential. The quantum situation is different. Suppose that we are certain the particle is at a given time t_{1} inside the interval [−a, a]. This means that the support of its wavefunction ψ(x, t_{1}) is contained in [−a, a]. One can show that the time evolution is such that the wave function will spread instantaneously; at each later time t_{2} it will be nonvanishing at arbitrary points x outside [−a, a], i.e. there is a nonzero probability of finding the particle outside the walls. Conversely, if we know the particle to be outside the walls at an initial time, the chance that we may find it inside at a later time is nonzero. The reason for this is that a strictly localized wave function, i.e. a wave function with compact support, contains all possible energies in its decomposition with respect to the eigenfunctions of the energy operator. A wave function of which the components are cut off below a certain energy, say the energy E_{0} of the potential walls, is not localized. The corresponding particle can be found inside as well outside a given interval. This phenomenon that a quantum particle cannot be contained in a region for an indefinite lenght of time and it can always pass through potential walls is called the tunnel effect .

9. THE HARMONIC OSCILLATOR

9.1. Introduction The great success of quantum mechanics between the years 1924 and 1927 was its ability to give a qualitatively and quantitatively satisfactory description of atomic spectra. By this it established itself as the theoretical model for submicroscopic physics. A simple but typical example in this context was the hydrogen atom . An even simpler quantum mechanical model is that of the harmonic oscillator , which will be treated in this chapter. Like the model for the hydrogen atom it can be explicitly solved by elementary means, i.e. the formulas for its energy levels and energy eigenstates can be found in closed form. The differences between the quantum version and the classical version of the harmonic oscillator are very characteristic for the differences between quantum and classical mechanics in general. This makes it excellent for pedagogical purposes. It is also of great practical use, because it gives a good first approximation to many more complicated and more realistic physical systems.

9.2. The classical harmonic oscillator A well-known system in elementary classical physics is that of a particle attached to one end of a metal spring. As long as the spring is not stretched too much, the particle will be driven back, in the direction of the point where the string is fixed, with a force proportional to distance to this point. This situation is idealized in the mechanical model of a classical harmonic oscillator : a point particle of mass m moving in R^{1} under the influence of a force F = −kx, with k the ‘spring constant’, implying that Newton’s equation F = ma = m\frac{d^{2}x}{dt^{2}} is

\frac{d^{2}}{dt^{2}} x(t) + \frac{k}{m}x(t) = 0.

Note that the force on the particle comes from the potential V (x) = \frac{1}{2}kx^{2}. The general solution, for arbitrary initial position x(0) and velocity ẋ(0) = (\frac{dx}{dt})_{t=0} is easily found to be

x(t) = ẋ(0)ω^{−1} sin ωt + x(0) cos ωt,

with ω defined as \sqrt{k/m}. The motion is periodic ; the particle oscillates around the origin x = 0 with ‘circle frequency’ ω, a characteristic constant of the system. Kinetic energy is constantly transformed into potential energy and vice versa, but the total energy remains the same. This energy depends depends on the initial values of the motion or on the maximum amplitude |x_{max}| = (2E/mω)^{1/2}. The possible energy values of the system form a continuum ; E can have an arbitrary nonnegative value, with for E = 0 the particle at rest in the origin x = 0, as a limiting case, the only stationary state.

The system can be put in Hamiltonian form, with canonical coordinates p and x, with a Hamiltonian function

H(p, x) = \frac{p^{2}}{2m} + \frac{1}{2} m ω^{2}x^{2}

and Hamiltonian equations

\frac{dx}{dt} = \frac{p}{m}, \frac{dp}{dt} = −m ω^{2}x,

together equivalent to the second order equation

\frac{d^{2}x}{dt^{2}} + ω^{2}x = 0.

9.3. The quantum oscillator As was already indicated in the introductory section, many important systems in physics can in first approximation be desribed by a harmonic oscilator or by systems of harmonic oscillators. The basic reason for this is that a general potential V (x) with a minimum in x = 0 can be expanded around x = 0 as

V (x) = V (0) + V′(0)x + \frac{1}{2}V′′(0)x^{2} + …

with V′, V′′ the first, respectively the second derivative of V. The first term is a constant and does therefore not affect the dynamics, the second term vanishes because V has a minimum in x = 0, so the first nonzero term is a term quadratic in x, i.e. a harmonic potential. A similar argument can of course be used when V has a minimum in an arbitrary x = x_{0}. The role of the harmonic oscillator model, as a simple starting point for the description of situations in which particles are bound to each other by strong forces, is even more important in quantum physics than in classical physics. This gives another justification for the detailed discussion of the quantum mechanical 1-dimensional harmonic oscillator in this chapter.

The state space H for the quantum mechanical oscillator is the Hilbert space L^{2}(R^{1}, dx), its Hamiltonian operator

H = \frac{1}{2m}P^{2} + \frac{1}{2}mω^{2}Q^{2}.

This operator can be shown to be essentially selfadjoint on a suitable domain of smooth functions, e.g. S(R^{1}).

Remark : The ‘prescription’ by which the quantum Hamiltonian is obtained from the classical Hamiltonian is here still unambiguous, even though P does not commute with Q. There would be a problem if the classical Hamiltonian function would for instance contain a term p^{3}q^{3}. The operator P^{3}Q^{3} is not hermitian symmetric, so is not admissable. Classically one has

p^{3}q^{3} = \frac{1}{2} ((pq)^{3} + (qp)^{3}) = \frac{1}{2} (p^{3}q^{3} + q^{3}p^{3}) .

Both the combinations \frac{1}{2} ((P Q)^{3} + (QP)^{3}) and \frac{1}{2} (P^{3}Q^{3} + Q^{3}P^{3}) are hermitian symmetric. They are however not the same, so it is not clear which one should be chosen as quantum observable. It also not immediately obvious whether both operators are essentially selfadjoint.

Studying a quantum system the central task is to find the eigenvalues and eigenvectors, or more rigorously the spectral resolution, of the Hamiltonian – usually the energy operator – of the system. Only in a few cases this problem can be solved exactly. The hydrogen atom is one of these cases; the harmonic oscillator another.

Solving the eigenvalue / eigenvector problem Hψ = Eψ for the harmonic oscillator means solving the differential equation

(−\frac{h̄}{2m} \frac{d^{2}}{dx^{2}} + \frac{1}{2}m ω^{2}x^{2}) ψ(x) = Eψ(x),

for suitable, i.e. square integrable functions ψ. This problem was solved first by a straightforward application of a general method for studying solutions of second order differential equations of a certain type; later a more algebraic method was found which uses in an ingenious way the quantum mechanical commutation relations. The first method will be briefly scketched here; the second will be treated in more detail in the next section.

Choose a new dimensionless variable

u = \sqrt{\frac{mω}{h̄}} x,

use instead of E the dimensionless quantity

ε = \frac{E}{h̄ω} − \frac{1}{2},

and define

f(u) = e^{\frac{mω}{2h̄} x^{2}}ψ(\sqrt{\frac{h̄}{mω}} x).

This leads to a differential equation for f as a function of u, which is equivalent to the original equation and which is

\frac{d^{2}f}{du^{2}} − 2u\frac{df}{du} + 2εf = 0.

Expand the function f into a power series f(u) = a_{0} + a_{1}u + a_{2}u^{2} + . . .. Substitution of this series in the differential equation gives a simple recursion relation for the coefficients,

a_{n+2} = \frac{2(n − ε)}{(n + 1)(n + 2)}a_{n},

for n = 0, 1, 2, . . .. Given a_{0}, this relation determines the coefficients for even n; given a_{1} one obtains those for odd n. The first case gives an even function of the variable u, the second case an odd function. Note that substitution of the power series for ψ in the original equation would not give a useful recursion relation; in fact the nontrivial part of this procedure is to find a transformation to a new equation for which the recursion relation is simple and determines the coefficients. One next studies the asymptotic behaviour of the coefficients a_{n} for n → ∞ and finds that there are two possibilities. The first is that series of coefficients {a_{n}} is infinite. In that case the function f(u) behaves as ce^{u^{2}} for u → ±∞. This gives for the original function ψ a total exponential factor e^{\frac{1}{2} u^{2}}. Such a solution ψ is not square integrable and is therefore not acceptable as wave function. We are left with the second possibility: the series breaks off and f(u) is a polynomial. This happens precisely for integer values of ε. For ε = n_{0} we clearly have that all a_{n} are nonzero for n ≤ n_{0} and vanish for n ≥ n_{0} + 2. These results together give the solution of the original eigenvalue / eigenvector problem Hψ = Eψ as

E_{n} = h̄ω(n + 1/2)

for the eigenvalues, and for the eigenfunctions

ψ_{n}(x) = N_{n}H_{n}(\sqrt{mω/h̄} x) e^{− \frac{mω}{2h̄} x^{2}},

for n = 0, 1, 2, . . .. The N_{n} are normalization constants equal to

N_{n} = \frac{1}{2^{n/2}(n!)^{1/2}} (\frac{mω}{h̄π} )^{1/4}.

The H_{n}(·) are the Hermite polynomials , well-known from other areas of applied mathematics. The first five Hermite polynomials are

H_{0}(u) = 1
H_{1}(u) = 2u
H_{2}(u) = −2 + 4u^{2}
H_{3}(u) = −12u + 8u^{3}
H_{4}(u) = 12 − 48u^{2} + 16u^{4}.

One finally shows that this is the complete solution: the spectrum of H is purely discrete and the eigenfunctions ψ_{n} form an orthonormal basis in the Hilbert space H.

These results should be compared with what we know about the classical situation: A classical oscillator has a continuum of possible energy values, the interval [0, +∞) and a single stationary state, the situation in which the particle has energy 0 and is at rest in the origin. The possible energy values of a quantum oscillator are discrete , E_{n} = h̄ω(n + 1/2), for n = 0, 1, …; there is an infinite sequence of stationary states, the corresponding eigenstates ψ_{n}, in which the ‘groundstate’ ψ_{0}, the state with the lowest energy, has a nonzero ‘rest energy’ \frac{1}{2}h̄ω.

9.4. Lowering and raising operators The starting point of the second method is a transformation of the operators P and Q into a pair of non-selfadjoint operators A and A^{∗}. Define

A = αP + iβQ, A^{∗} = αP − iβQ.

By choosing the real parameters α and β as

α = (2h̄)^{− \frac{1}{2}} (mω)^{− \frac{1}{2}} , β = −(2h̄)^{− \frac{1}{2}} (mω)^{\frac{1}{2}}

and using the basic commutation relation

[P, Q] = \frac{h̄}{i} 1

one finds for the new operators the commutation relation

[A, A^{∗}] = 1

and for the Hamiltonian operator H the expression

H = h̄ω(A^{∗}A + 1/2).

In all this we use S(R^{1}) as common invariant domain of definition for the operators. Denote A^{∗}A as N. The operators N and H have the same eigenvectors; eigenvalues ε of N correspond with eigenvalues E of H through the relation E = h̄ω(ε + 1/2). We therefore study the slightly more convenient eigenvalue / eigenvector problem

Nψ = εψ.

The solution of this problem is obtained in a procedure consisting of the following steps:

a. From the basic commutation relation [A, A^{∗}] = 1 one easily derives the relations

[N, A] = −A, [N, A^{∗}] = A^{∗}.

(For example: [N, A] = A^{∗}AA − AA^{∗}A = −[A, A^{∗}]A = −A).

b. These commutation relations are used to prove a lemma, which is the principal ingredient of the procedure: Lemma : If ψ is an eigenvector of N with eigenvalue ε, then either Aψ = 0 or the vector Aψ is an eigenvector of N with eigenvalue ε − 1. At the same time either A^{∗}ψ = 0 or the vector A^{∗}ψ is an eigenvector of N with eigenvalue ε + 1. Proof : N(Aψ) = [N, A]ψ + ANψ = −Aψ + εAψ = (ε − 1)(Aψ), which states that Aψ is an eigenvector of N, with eigenvalue ε − 1, unless Aψ = 0. Similarly one has N(A^{∗}ψ) = [N, A^{∗}]ψ + A^{∗}Nψ = A^{∗}ψ + εA^{∗}ψ = (ε + 1)(A^{∗}ψ). Remark : Because of these properties A is called a lowering operator and A^{∗} a raising operator . Systems of such lowering and raising operators A_{1}, A_{2}, … and A_{1}^{∗}, A_{2}^{∗}, …, with commutation relations

[A_{j}, A_{k}^{∗}] = δ_{jk}1, [A_{j}, A_{k}] = [A_{j}^{∗}, A_{k}^{∗}] = 0,

are important in other areas of physics. In elementary particle physics such operators are called annihilation and creation operators .

c. Eigenvalues of N are necessarily nonnegative. Let ε be an eigenvalue, with ψ the corrresponding eigenvector. One has

0 ≤ (Aψ, Aψ) = (ψ, A^{∗}Aψ) = (ψ, Nψ) = ε(ψ, ψ).

Because (ψ, ψ) > 0, ε must be nonnegative .

d. Suppose there is an eigenvector ψ with eigenvalue ε. Repeated action of A on ψ will give a sequence of new eigenvectors Aψ, A^{2}ψ, … with eigenvalues ε − 1, ε − 2, . . .. These eigenvalues cannot become negative, so this sequence has to break off. This means that that there is a vector ψ_{0} = 0 and a nonegative real number ε_{0}, such that Nψ_{0} = ε_{0}ψ_{0} and Aψ_{0} = 0. This implies 0 = (Aψ_{0}, Aψ_{0}) = (ψ_{0}, Nψ_{0}) = ε_{0}(ψ_{0}, ψ_{0}). Because (ψ_{0}, ψ_{0}) > 0 this gives ε_{0} = 0. The result of this is that the existence of an eigenvector ψ with eigenvalue ε entails the existence of an eigenvector ψ_{0} with a lowest eigenvalue ε_{0} = 0. Starting from this ψ_{0} we can apply the operator A^{∗}, and obtain a sequence of eigenvectors A^{∗}ψ_{0}, (A^{∗})^{2}ψ_{0}, …, with eigenvalues 1, 2, . . .. Suppose that for a given n from [0, 1, 2, …] the vector (A^{∗})^{n+1}ψ_{0} = 0, while (A^{∗})^{n} ≠ 0. Then

((A^{∗})^{n+1}ψ_{0}, (A^{∗})^{n+1}ψ_{0}) = ((A^{∗})^{n}ψ_{0}, AA^{∗}(A^{∗})^{n}ψ) =
= ((A^{∗})^{n}ψ_{0}, (N + 1)(A^{∗})^{n}ψ_{0}) = (n + 1)((A^{∗})^{n}ψ_{0}, (A^{∗})^{n}ψ_{0}) ≠ 0.

This means that the upward sequence ψ_{0}, A^{∗}ψ_{0}, (A^{∗})^{2}ψ_{0}, … does not break off. Note that the action of A and A^{∗} leaves this sequence invariant, up to real factors: A((A^{∗})^{n}ψ_{0}) = (AA^{∗})((A^{∗})^{n−1}ψ_{0}) = (1 + N)((A^{∗})^{n−1}ψ_{0}) = (n + 1)(A^{∗})^{n−1}ψ_{0} and A^{∗}((A^{∗})^{n}ψ_{0}) = (A^{∗})^{n+1}ψ_{0}. The vectors (A^{∗})^{n}ψ can be normalized by using the above formula. to normalize the. One has

||(A^{∗})^{n+1}ψ_{0}||^{2} = ((A^{∗})^{n+1}ψ_{0}, (A^{∗})^{n+1}ψ_{0}) = (n + 1)((A^{∗})^{n}ψ_{0}, (A^{∗})^{n}ψ_{0}) =

= (n + 1) ||(A^{∗})^{n}ψ_{0}||^{2} = … = (n + 1)! ||ψ_{0}||^{2}.

We choose (ψ_{O}, ψ_{0}) = 1 and obtain then an orthonormal sequence of eigenvectors ψ_{0}, ψ_{1}, …, with ψ_{n} defined as

ψ_{n} = \frac{1}{\sqrt{n!}} (A^{∗})^{n}ψ_{0}.

e. Our result sofar is that the eigenvectors of N, in the domain S(R^{1}), occur in infinite orthonormal sequences ψ_{0}, ψ_{1}, ψ_{2}, …, with corresponding sequences of eigenvalues ε = 0, 1, 2, . . .. The number of such sequences depends on the number of (linear independent) ‘ground states’ ψ_{0}. To find the possible vectors ψ_{0} one must leave the purely algebraic context and return to the the explicit Hilbert space picture. A state ψ_{0} with lowest eigenvalue ε_{0} = 0 satisfies Aψ_{0} = 0, i.e. is a solution of the equation (αP + iβ)ψ_{0} = 0, or more explicitly,

\frac{d}{dx}ψ_{0}(x) = \frac{β}{αh̄}xψ_{0}(x).

This is an elementary differential equation, which has the general solution

ψ_{0}(x) = c e^{\frac{β}{2αh̄} x^{2}},

with c a constant. Substitution of the expressions for α and β gives

ψ_{0}(x) = c e^{− \frac{mω}{2h̄} x^{2}}.

The standard Gaussian integral ∫_{−∞}^{+∞} e^{−y^{2}}dy = π^{1/2} is used to normalize ψ_{0}, which gives for the constant c the condition |c|^{2} = (mω/h̄π)^{1/2}. With the obvious choice for c one obtains finally the single ‘ground state’ eigenfunction

ψ_{0}(x) = (\frac{mω}{h̄π} )^{\frac{1}{4}} e^{− \frac{mω}{2h̄} x^{2}}.

From this the other eigenfunctions ψ_{n} are obtained as

ψ_{n}(x) = \frac{1}{\sqrt{n!}} ((A^{∗})^{n}ψ_{0})(x).

One can show, using explicit Hilbert space properties, that the system {ψ_{n}} forms an orthonormal basis for H = L^{2}(R^{1}, dx). This shows that the ψ_{n}, with their eigenvalues ε = n, give a complete (discrete) spectral resolution for the operator N and therefore for the operator H = h̄ω(N + 1/2), with eigenvalues E_{n} = h̄ω(n + 1/2) . One can of course verify that the ψ_{n} here are the same as the ones found in the preceding section.

9.5. Time evolution The quantum oscillator has an infinite sequence of stationary states, the energy eigenfunctions ψ_{0}, ψ_{1}, …, for which the predictions of the results of measurements of arbitrary observables are constant in time. (Note however that an actual measurement would instantly transform the stationary state into an in general nonstationary state, in the case of a measurement of position with a wave function concentrated around the observed position value). In Section 8.1. we derived equations for the time evolution of the mean values of position and momentum for a general wave packet, which strongly resemble the classical equations. For the harmonic oscillator these are the classical equations: because of \overline{\frac{dV}{dx}} = \overline{\frac{d}{dx}(\frac{1}{2}mω^{2})x^{2}} = mω^{2}\overline{x} one obtains

m\frac{d}{dt}\overline{x}(t) = \overline{p}(t)

\frac{d}{dt}\overline{p}(t) = − mω^{2}\overline{x}(t).

This implies that all the stationary states ψ_{n} have \overline{x} = 0 and \overline{p} = 0, just like the single stationary state x = 0, p = 0 for the classical oscillator, but at the same time with Δx ≠ 0 and Δx ≠ 0. The ground state ψ_{0} is a minimal uncertainty state , as will be shown in the next section. A general state ψ can be expanded as ψ = ∑_{n=0}^{∞} c_{n}ψ; its time evolution is therefore given by the formula

ψ(t) = ∑_{n=0}^{∞} e^{−iω(n+1/2)t}c_{n}(0)ψ_{n}.

The time evolution is particularly interesting for the so called coherent states , a class of minimal uncertainty states which emerge here quite naturally by using the operators A and A^{∗}. This will be discussed in the next section.

9.6. Coherent states The ground state ψ is a solution of the equation Aψ = 0, i.e. it is an eigenvector of the operator A with eigenvalue 0; and as such it is unique up to scalar factor. Has A has other eigenvectors? The eigenvalue / eigenvector problem for A is an unusual one, because A is not symmetric or selfadjoint. It has nevertheless a fairly simple solution which is moreover of physical interest. To find this solution one needs the commutation relation [A, (A^{∗})^{n}] = n(A^{∗})^{n−1} which is obtained from the basic relation [A, A^{∗}] = 1 by induction. Expansion of the left-hand side of the equation Aϕ_{λ} = λϕ_{λ} with respect to the orthonormal basis ψ_{0}, ψ_{1}, … gives

Aϕ_{λ} = A (∑_{n=0}^{∞} c_{n}ψ_{n}) = ∑_{n=0}^{∞} \frac{c_{n}}{\sqrt{n!}}A(A^{∗})^{n}ψ_{0} =

= ∑_{n=0}^{∞} \frac{c_{n}}{\sqrt{n!}} ([A, (A^{∗})^{n}] + (A^{∗})^{n}A) ψ_{0} =

= ∑_{n=0}^{∞} \frac{c_{n}\sqrt{n}}{\sqrt{(n − 1)!}}(A^{∗})^{n−1}ψ_{0} =

= ∑_{n=1}^{∞} c_{n}\sqrt{n} ψ_{n−1} = ∑_{n=0}^{∞} c_{n+1}\sqrt{n + 1} ψ_{n}.

Note that this is only well-defined if the coefficients c_{n+1}\sqrt{n + 1} are such that

∑_{n=0}^{∞} |c_{n}|^{2} \frac{|λ|^{2n}}{n!} < ∞.

Expansion of the right-hand side gives

λϕ_{λ} = ∑_{n=0}^{∞} λc_{n} ψ_{n}.

By comparing the two expansions one obtains the simple recursion relation c_{n+1}\sqrt{n + 1} = λc_{n}, which, after choosing c_{0} = 1, gives c_{n} = λ^{n}/\sqrt{n!}, for n = 0, 1, . . .. The series of coefficients λ^{n}/\sqrt{n!} is l_{2}-summable, so the vector ϕ_{λ} is well-defined. One has

||ϕ_{λ}||^{2} = ∑_{n=0}^{∞} \frac{|λ|^{2n}}{n!} = e^{|λ|^{2}},

and similarly, for two different complex numbers λ_{1} and λ_{2},

(ϕ_{λ_{1}}, ϕ_{λ_{2}}) = ∑_{n=0}^{∞} \frac{(\overline{λ_{1}}λ_{2})^{n}}{n!} = e^{\overline{λ_{1}}λ_{2}}.

The vector ϕ_{λ} can be written as

ϕ_{λ} = ∑_{n=0}^{∞} \frac{λ^{n}}{\sqrt{n!}}ψ_{n} = ∑_{n=0}^{∞} \frac{λ^{n}}{n!} (A^{∗})^{n}ψ_{0} = e^{λA^{∗}}ψ_{0},

a very appealing and convenient expression, although one should realize that this does not mean that the operator-valued exponential function e^{λA^{∗}} has in itself a good mathematical meaning, but only that the special sequence of vectors

ψ_{0} + A^{∗}ψ_{0} + \frac{1}{2!}(A^{∗})^{2}ψ_{0} + \frac{1}{3!}(A^{∗})^{3}ψ_{0} + …

is convergent. In this manner we have found a complete solution of the rather unusual eigenvalue / eigenvector problem Aϕ_{λ} = λϕ_{λ}: for a each complex number λ there is a vector

ϕ_{λ} = ∑_{n=0}^{∞} \frac{λ^{n}}{\sqrt{n!}}ψ_{n} = e^{λA^{∗}}ψ_{0}.

The vectors ϕ_{λ} are normalizable, with norm ||ϕ_{n}|| = e^{|λ|}, and inner product (ϕ_{λ_{1}}, ϕ_{λ_{2}}) = e^{\overline{λ_{1}}λ_{2}}. The eigenvalues λ form a continuum, the complex plane, but the eigenvectors are nevertheless normalizable. The uncountable system {ϕ_{λ}}_{λ∈C} is linearly independent in the sense that each finite subset is linearly independent, even though the Hilbert space is separable. It is a basis of the Hilbert space H, not in the usual sense, but in a generalized ‘continuous’ way. An arbitrary vector ψ in H can be written as

ψ = \frac{1}{2πi} ∫_{C} F_{ψ}(λ) e^{−|λ|^{2}}d\overline{λ}dλ,

with \frac{1}{2πi}d\overline{λ}dλ = \frac{1}{π} dxdy, for λ = x+iy, and with F_{ψ} a function which is uniquely determined by

F_{ψ}(λ) = (ϕ_{λ}, ψ).

One can show that F_{ψ}(λ) is anti-holomorphic , i.e. its complex conjugate \overline{F_{ψ}(λ)} is holomorphic, and that it satisfies the growth condition

∫_{C} |F_{ψ}(λ)|^{2}e^{−|λ|^{2}}d\overline{λ}dλ < ∞.

Such functions form a Hilbert space Ĥ with respect to the inner product

(F_{1}, F_{2}) = \frac{1}{2πi} ∫_{C} \overline{F_{1}(λ)}F_{2}(λ) e^{−|λ|^{2}}d\overline{λ}dλ.

The relation between the vectors ψ in H and the anti-holomorphic functions F in Ĥ given above is a unitary transformation between H and Ĥ. The space Ĥ is called the Bargmann Hilbert space of anti-holomorphic functions – or holomorphic functions, if one uses a slightly different convention. It has many interesting mathematical properties.

Remark : Note the difference between this notion of a ‘continuous base’ and the one suggested by the system of nonnormalizable eigenfunctions ϕ_{p}(x) = c e^{\frac{i}{h̄} px} of the momentum operator P.

In this quantum mechanical context the vectors ϕ_{λ}, or their normalized versions

ϕ̃_{λ} = e^{− \frac{1}{2} |λ|^{2}}ϕ_{λ},

are called coherent states , a term which has its origin in an application in quantum optics, a subject that cannot be discussed here. (Strictly speaking the term ‘state’ should be reserved for the normalized vectors ϕ̃_{λ}, but the ϕ_{λ} are often more convenient to use and are therefore usually also called states.)

To understand the physical properties of (normalized) coherent states, we calculate the expectation values \overline{p}, \overline{x} and standard deviations Δp, Δx. The inverse transformation formulas

P = \frac{1}{2α}(A + A^{∗})

Q = \frac{1}{2iβ} (A − A^{∗}),

with α = (2mh̄ω)^{−1/2} and β = −(mω/2h̄)^{1/2}, allow us to express P and Q in A and A^{∗}. Note that one has

(ϕ̃_{λ}, (A^{∗})^{m}A^{n}ϕ̃_{λ}) = (A^{m}ϕ̃_{λ}, A^{n}ϕ̃_{λ}) = \overline{λ}^{m}λ^{n},

for all m, n = 0, 1, . . .. An arbitrary algebraic expression in the A and A^{∗} can be written in what is called normally ordered form i.e. as a sum of such double monomials, if necessary after a rearrangement using the basic commutation relation [A, A^{∗}] = 1. With this one obtains

\overline{p} = (ϕ̃_{λ}, P ϕ̃_{λ}) = \frac{1}{2α}(ϕ̃_{λ}, (A + A^{∗})ϕ̃_{λ}) = \frac{1}{2α}(λ + \overline{λ}) = \frac{1}{α} Re λ,

and similarly

\overline{x} = (ϕ̃_{λ}, Qϕ̃_{λ}) = \frac{1}{2iβ} (λ − \overline{λ}) = \frac{1}{β} Im λ.

Together this gives

λ = α \overline{p} + iβ \overline{x}.

For the calculation of Δp one needs

\overline{p^{2}} = (ϕ̃_{λ}, P^{2}ϕ̃_{λ}) = \frac{1}{4α^{2}} (ϕ̃_{λ}, (A^{2} + (A^{∗})^{2} + A^{∗}A + AA^{∗})ϕ̃_{λ}) =

= \frac{1}{4α^{2}} (ϕ̃_{λ}, (A^{2} + (A^{∗})^{2} + 2A^{∗}A + 1)ϕ̃_{λ}) = \frac{1}{4α^{2}} (λ^{2} + \overline{λ}^{2} + 2|λ|^{2} + 1).

One finds

(Δp)^{2} = \overline{p^{2}} − (\overline{p})^{2} = \frac{1}{4α^{2}} (λ^{2} + \overline{λ}^{2} + 2|λ|^{2} + 1) − \frac{1}{4α^{2}} (λ + \overline{λ})^{2} = \frac{1}{4α^{2}} ,

and therefore Δp = 1/2α = (mω)^{1/2}(h̄/2)^{1/2}. Similarly one obtains (Δx)^{2} = 1/4β^{2} or Δx = −1/2β = (mω)^{−1/2}(h̄/2)^{1/2}. Combining the results on p and x one obtains finally

Δp Δx = −\frac{1}{4αβ} = \frac{1}{2}h̄,

which means that each normalized coherent state ϕ̃_{λ} is a minimum uncertainty state . Conclusion : For each value \overline{p} and \overline{x} there is a minimum uncertainty state, the normalized coherent state ϕ̃_{λ}, an eigenvector of A with eigenvalue λ, with

λ = α \overline{p} + iβ \overline{x} = (2mh̄ω)^{1/2}\overline{p} + i(mω/2h̄)^{1/2}\overline{x}.

All these minimum states have the same standard deviations Δp = 1/2α = (mω)^{1/2}(h̄/2)^{1/2} and Δx = −1/2β = (mω)^{−1/2}(h̄/2)^{1/2}. The minimum states associated with a quantum harmonic oscillator in this way form a 2-parameter subset of the 3-parameter set of minimum states found in Section 7.4.

9.7. Time evolution of coherent states An interesting physical aspect of the coherent states of an harmonic oscillator is their behaviour under time evolution. For an arbitrary state ψ = ∑_{n=0}^{∞} c_{n}ψ_{n} the time evolution is given by the 1-parameter group of unitary operators e^{− \frac{i}{h̄} tH}, acting as

e^{− \frac{i}{h̄} tH}ψ = ∑_{n=0}^{∞} e^{−iω(n+1/2)t}c_{n}ψ_{n}.

For a coherent state

ϕ_{λ} = ∑_{n=0}^{∞} \frac{λ^{n}}{\sqrt{n!}}ψ_{n}

the time evolution becomes

e^{− \frac{i}{h̄} tH}ϕ_{λ} = ∑_{n=0}^{∞} \frac{e^{−iω(n+1/2)t}λ^{n}}{\sqrt{n!}} ψ_{n} = e^{− \frac{i}{2} ωt} ∑_{n=0}^{∞} \begin{matrix}λ(t) & n \\ √ & \overline{n!} \end{matrix} ψ_{n},

with λ(t) = e^{−iωt}λ. This means that the coherent state ϕ_{λ} is transformed – up to the phase factor e^{− \frac{i}{2} ωt} – into a new coherent state ϕ_{λ(t)}: the set of coherent states is mapped onto itself.

The wave function of a (normalized) coherent state ϕ̃_{λ} moves periodically, up to an irrelevant phase factor. The corresponding probability density |ϕ̃_{λ}(x)|^{2} is Gaussian. It moves exactly like a ‘smeared’ classical oscillator with frequency ω. Its width is constant in time and equal to (mω)^{−1/2}(h̄/2)^{1/2}. In the classical limit, the limit of a macroscopic situation, this becomes the harmonic motion of a point particle.