Topics in Advanced Quantum Mechanics

Index

CHAPTER I

PROPAGATOR METHODS

I.1 BASIC QUANTUM MECHANICS

(0)〉 at time t = 0, what is the state at a later time t (t)〉? The answer is provided by the Schrödinger equation

where Ĥ is the Hamiltonian operator.† Usually one sees this equation expressed in terms of the coordinate space projection of the state vector — i.e. (x, t) where††

The time-evolution of the wavefunction is then given by

In order to evaluate the matrix element on the right we can insert a complete set of co-ordinate states

yielding

. In general, the Hamiltonian Ĥ can be written in terms of kinetic and potential energy components as

Here

so

† Note that thoughout this book, we set ħ=1.

†† For simplicity of notation, we shall work here in one dimension. However, generalization to three dimensions is obvious.

In order to represent the kinetic energy piece we can insert a complete set of momentum states such that

. Then

yielding

Since 〈x|p〉 is simply a plane wave we have

we have

Substitution back into Eq. 1.3 yields

which is the usual version of the Schrodinger equation, where

provides the representation of the operator Ĥ in coordinate space. For a free particle this reduces to the simple form

Time Development Operator

An alternative formulation of this problem is in terms of the time development operator Û (t, t′) defined via

with the boundary condition

For the case of a free particle, obeying

the solution for Û(0) (t, 0) is

where

is the usual theta function. For example, if

we find

Although one could straightforwardly evaluate this power series, it is easier to note the identity [Bl 68]

Then using

we find

We note that

which obviously exhibits the canonical spreading experienced by such a wavepacket.

We can equivalently perform the above calculation in momentum space, where the time development operator has the simple form

If

we have

Then

We can return to coordinate space via

which agrees precisely with Eq. 1.24 found via coordinate space methods.

PROBLEM I.1.1

Wave Packet Spreading: A Paradox

It was demonstrated above using the identity

that a Gaussian wavepacket

evolves in time via

where

Then

where

i) Show that

so that the wavepacket remains normalized to unity but has a width

which evolves with time. This is simply the usual spreading of a quantum mechanical wave packet.

ii) Derive the time evolution of the Gaussian wavepacket without exploiting the identity by using a power series expansion

iii) Now suppose that

where N is a normalization constant. Although this functional form may look a bit strange, a little thought should convince one that the wavefunction and all its derivatives are continuous at any point on the real line. However, it is easy to see that

vanishes for all time if |x| ≥ a since

Hence, this type of wavepacket apparently does not undergo spreading. Is this assertion correct? If not, where have we made an error in our analysis and what does the actual time evolved wavefunction look like [HoS 72]?

I.2 THE PROPAGATOR

One can evaluate the co-ordinate space matrix element of the time development operator by transforming to momentum space and back again.

DF is usually called the propagator, as it gives the amplitude for a particle produced at position x at time 0 to propagate to position x′ at time t.

Just as a check we can verify that this form of the propagator does indeed generate the time development of the freely moving Gaussian wavefunction

in complete agreement with expression derived in sect. I.1

Path Integrals and the Propagator

Before going further, it is useful to note an alternative way by which the propagator can be calculated—the Feynman path integral [FeH 65]

where the notation is that the integral represents a sum over all paths x(t) connecting the initial and final spacetime points — x, 0 and x′, t is the classical action associated with that path. The path integration can be carried out by

Fig. I.1: A particular time slice used in calculation of the propagator.

dividing the time interval 0 – t into n . This provides a set of times ti spaced a distance e apart between the values 0 and t. At each time ti we select a point xi. A path is constructed by connecting all possible xi points so selected by straight lines as shown in Figure I.1 and the path integral is written (setting ħ = 1) as

where A is a normalization constant which defines the measure— note that there is one factor of A → 0 we can evaluate the action for each line segment in the infinitesimal approximation. For the free particle we have

The integrations may be performed sequentially

yielding

The constant A may be determined by use of the completeness condition

If we pick t <<<1 then

is small, the exponential will rapidly oscillate and thereby wash out the integral unless x ≅ x′. Thus, we can write

→ 0 we must pick

so that the free propagator becomes, using t = n

in complete agreement with the expression derived via more conventional means (cf. Eq. 2.1).

The reason that the propagator can be written as a path integral can be understood by using the completeness relation

For later use, we shall give the derivation here for the general case involving interaction with a potential V . Starting with the definition

and breaking the time interval tf – ti (assumed to be positive) into n discrete steps of size

we can write

In the limit of large n the time slices become infinitesimal and

Introducing a complete set of momentum states, we have

and, taking the continuum limit, we find the path integral prescription

where

is the classical action.

Classical Connection

Perhaps the most peculiar and fascinating aspect of this prescription is that all paths connecting the spacetime endpoints must be included in the summation. This appears to be in total contradiction with the classical mechanics result that a particle traverses a well-defined trajectory. The resolution of this apparent paradox may be found by explicitly restoring the dependence on ħ and noting that the path integral prescription is given by

Classical physics results as ħ → 0, and in this limit a slight change in the path x(t(t) for which the action is stationary—i.e., Hamilton’s principle

In order to find such a path we take

integrate by parts and use the feature that the endpoints of the path are fixed, i.e., δx(0)= δx(t)= 0. Then

so that the trajectory which satisfies the stationary phase condition for arbitrary δx(t′) must obey

which is just the classical mechanics prescription for the motion of a freely moving particle, i.e., . In the limit ħ → 0 the classical trajectory represents the only path contributing to the path integral and the paradox is resolved.

One can also get a feel for the meaning of the propagator by noting that since

if we take

so that at t = 0 the particle is located precisely at the origin, then

(x, t; 0, 0) is just the Schrödinger wavefunction of a freely moving particle which started at the origin at time zero. If we look at a specific location x0, t0 we would say classically that if a particle is observed at this point then it must have momentum

and energy

Examining the variation of the phase of the wavefunction in the vicinity of x0, t0, we find

Thus in the vicinity of this point we can write

so that both the wavelength associated with the particle

and the corresponding frequency

are given by the usual quantum mechanical relations.

Finally, the probability that the particle is located between x and x + dx at time t is

and is independent of x. All momenta then are equally likely at t = 0, as would be expected from the momentum space representation of the co-ordinate space wavefunction

. We conclude that all our intuitive notions are satisfied by the propagator, Eq. 2.27.

Fig. I.2: When t > 0 the contour is closed by means of a large semicircle in the lower half plane.

Frequency Space Representation

Before moving on to the more interesting case of motion in the presence of a potential, it is important to note that the time development operator is often used in Fourier transform or frequency space form rather than in its time representation. Before examining this result, however, it is useful to prove a simple mathematical identity. Consider the integral

If t > 0 the contour can be closed in the lower half plane by means of a large semicircle (cf. . The integral can then be evaluated by means of the residue theorem [MaW 64]. There exists a single pole at ω = a − i and the integral is found to be

On the other hand, if t < 0 exponential damping of the semicircular contribution demands that we close the contour in the upper half plane. In this case there is no singularity so

We have in general then

so that, replacing a , an alternative way to represent the free time development operator is

i.e., Û(0)(t, 0) can be written as a Fourier transform with

Other Representations

in terms either of its momentum space

or coordinate space matrix elements

Defining

we can explicitly evaluate the latter

If x – x′ > 0 we close the contour in the upper half plane and pick up the pole at p0 + i , yielding

while if x – x′ < 0 we must close the contour in the lower half plane and pick up the pole at –p0 − i , yielding

The general result can be written as

and will be useful later.

PROBLEM I.2.1

The Hamiltonian Path Integral

An alternative—Hamiltonian—version of the path integral is often useful when one is dealing with non-Cartesian variables or with constrained systems.

i) Show that

ii) Using the result from i) demonstrate that the propagator may be written as

which is the form we were seeking. Note that here p, x are considered as independent variables.

I.3 HARMONIC OSCILLATOR PROPAGATOR

Having examined the form of the free propagator in the previous section, we now consider motion under the influence of a potential V . In this case the time development operator becomes (hereafter assuming t > 0)

which has the coordinate space representation

Provided that the Hamiltonian can be solved exactly to yield eigenvalues and eigenfunctions

we can find an exact representation of the propagator

(Note that the free particle propagator is of this form since

as before.) That Eq. 3.4 generates the time development of an arbitrary wavefunction is clear since, assuming t > 0

where

(x, 0) onto eigenstate ϕn (x). Eq. 3.6 then is simply the usual expansion of the wavefunction at later time t in terms of eigenstates of the Hamiltonian.

For soluble problems the propagator can generally be given simply and in closed form, and below we shall show how this is done for the case of the harmonic oscillator

However, before deriving the explicit form it is useful to review the solution of the harmonic oscillator problem using the technique due to Dirac [Di 58].

Harmonic Oscillator Review

We begin by defining the so called creation and annihilation operators

we find

or

Now look for eigenstates |n〉 such that

We can determine the properties of the eigenvalues n is hermitian—

—we have

i.e., n is real. Also n is non-negative since it can be written as the inner product of a state with itself.

Commutation relations are easily found

Also

with eigenvalue n – 1, since

is given by

so that

Similarly, operating repeatedly with â we can lower the eigenvalue even further

From Eq. 3.15, however, negative eigenvalues are not permitted, so we must eventually reach a state |1〉 such that

and

We conclude that the eigenvalue n must be an integer—n

with the normalization condition

then yields

Starting with the lowest energy (ground) state |0〉 we find

so that an arbitrary state |n〉 can be written as

The energy of the eigenstate |n〉 is given by

as expected.

, namely

The solution to this differential equation yields the familiar ground state wavefunction

. For the first excited state, we find

and one can generate the wavefunction of an arbitrary excited state via

where Hn (x) represents the Hermite polynomial of order n.

Harmonic Oscillator Propagator

We now return to the problem of the harmonic oscillator propagator. There exist a number of techniques by which this result may be obtained. For example, Itzykson and Zuber [ItZ 80] use a traditional time slice procedure in order to yield the closed form

(Note that Eq. 3.34 reduces to the free propagator result in the limit ω → 0.) However this procedure, while straightforward, is also lengthy and cumbersome. An alternate way to obtain the same result is by use of the Feynman path integral

but with the arbitrary trajectory x(t) characterized in terms of its deviation δx(t) from the classical path xcl (t), which satisfies

Then

where

Integrating the term linear in δx by parts, we find

where we have used the classical equation of motion and the fixed endpoint constraint— δx(t) = δx(0) = 0. We have then

and

i.e., the phase of the exponential is simply the action for the classical path! Writing

we require the boundary conditions

and the action is found to be

We must now eliminate A, ϕ in favor of x, x′. Noting that

and solving for cos ϕ we have

Thus

and

The time dependent prefactor DF (0, t; 0, 0) can be evaluated either via standard path integral techniques or via a shortcut. We first demonstrate the latter. Using the completeness property

Defining

we have, combining Eqs. 3.34 and 3.49

To simplify things pick x = x′ = 0. Then

whose solution is

in agreement with Eq. 3.34.

Determinant Methods and the Prefactor

A technique by which to evaluate the prefactor which is more generally useful is given below. Writing the path integral as

where δx(ti) = δx(tf) = 0 and integrating by parts we find

Now expand δx(t

where xn (t) satisfies

with xn (ti) = xf (tf) = 0 and is subject to the orthogonality condition

The sum over all possible trajectories can then be performed by summation over all expansion coefficients an

where N, N′ are normalization coefficients and

is the product of operator eigenvalues. For the harmonic oscillator

and the determinant can be evaluated using the identity [GrR 65]

i.e.,

where the constant is independent of ω, and may be absorbed into N′. The normalization constant N′ can in turn be determined by demanding that

i.e. that the free particle result be obtained in the limit ω → 0. Hence

which agrees with Eq. 3.53 found via the completeness property.

Wavefunction Connection

We can explicitly verify that Eq. 3.34 represents the correct form of the harmonic oscillator propagator by comparing with the sum over eigenstates, Eq. 3.4. We first examine the spectrum by taking the coordinate space trace

Using Eq. 3.34 we have

Thus

as expected. The wavefunctions may be obtained by expansion of the propagator itself:

Comparing with Eq. 3.4 we identify

etc., as the familiar harmonic oscillator wavefunctions.

PROBLEM I.3.1

Operator Solution of the Harmonic Oscillator

that

is a normalized eigenfunction of the harmonic oscillator.

Also, we know from elementary quantum mechanics that

where Hn (x) is a Hermite polynomial.

ii) Use these results to prove the recursion relation

iii) Use the recursion relation and H0 (z) = 1 to generate the first four Hermite polynomials.

PROBLEM I.3.2

Propagator for the Linear Potential

In order to gain experience dealing with the Feynman path integral, consider a particle of mass m moving in one dimension under the influence of a constant gravitational acceleration g. The corresponding potential energy is

and the Lagrangian becomes

The propagator for such a situation is exactly calculable and can be evaluated in at least two different ways:

i) Solve for the classical trajectory xcl (t) which obeys

and calculate the classical action

for a path satisfying the boundary conditions

Show that

Now imagine performing the path integral by expanding about this trajectory

Demonstrate that

Here

is the free propagator. However, we already know the form of the free propagator, yielding

Thus the propagator for the linear potential takes the form

ii) Evaluate the path integral by breaking the paths into infinitesimal time slices and performing successive integrations, i.e.

Demonstrate that the form after k integrations is

and that this expression reduces to the previously derived result in the limit that k → ∞.

PROBLEM 1.3.3

The Forced Harmonic Oscillator

An example of a problem which one can solve exactly to obtain the propagator is that of a harmonic oscillator which is acted upon by an external time varying force j(t). The corresponding Lagrangian is

i) Show that the classical solution to this problem which satisfies the boundary conditions

is given by