Michael Fowler, 10/14/07
Consider a macroscopic simple harmonic oscillator, and to keep things simple assume there are no interactions with the rest of the universe. We know how to describe the motion using classical mechanics: for a given initial position and momentum, classical mechanics correctly predicts the future path, as confirmed by experiments with real (admittedly not perfect) systems. But from the Hamiltonian we could also write down Schrödinger’s equation, and from that predict the future behavior of the system. Since we already know the answer from classical mechanics and experiment, quantum mechanics must give us the same result in the limiting case of a large system.
It is a worthwhile exercise to see just how this happens. Evidently, we cannot simply follow the classical method of specifying the initial position and momentum—the uncertainty principle won’t allow it. What we can do, though, is to take an initial state in which the position and momentum are specified as precisely as possible. Such a state is called a minimum uncertainty state (the details can be found in my earlier lecture on the Generalized Uncertainty Principle).
In fact, the ground state of a simple harmonic oscillator is a minimum uncertainty state. This is not too surprising—it’s just a localized wave packet centered at the origin. The system is as close to rest as possible, having only zero-point motion. What is surprising is that there are excited states of the pendulum in which this ground state wave packet swings backwards and forwards indefinitely, a quantum realization of the classical system, and the wave packet is always one of minimum uncertainty. Recall that this doesn’t happen for a free particle on a line—in that case, an initial minimal uncertainty wave packet spreads out because the different momentum components move at different speeds. But for the oscillator, the potential somehow keeps the wave packet together, a minimum uncertainty wave packet at all times. These remarkable quasi-classical states are called coherent states, and were discovered by Schrodinger himself. They are important in many quasi-classical contexts, including laser radiation.
Our task here is to construct and analyze these coherent states and to find how they relate to the usual energy eigenstates of the oscillator.
To define the notation, let us briefly recap the dynamics of the classical oscillator: the constant energy is
The classical motion is most simply described in phase space, a two-dimensional plot in the variables . In this space, the point corresponding to the position and momentum of the oscillator at an instant of time moves as time progresses at constant angular speed in a clockwise direction around the circle of radius centered at the origin.
(Note: phase space is usually defined in terms of the variables , but in describing the simple harmonic oscillator, the pair are more convenient.)
This motion is elegantly described by regarding the two-dimensional phase space as a complex plane, and defining the dimensionless complex variable
The time evolution in phase space is simply
The particular choice of (quantum!) scaling factor in defining z amounts to defining the unit of energy as , the natural unit for the oscillator: it is easy to check that if the classical energy then the dimensionless is simply the number (which is of course very large, so the ½ is insignificant).
We established in the lecture on the Generalized Uncertainty Principle that any minimum uncertainty one-dimensional wave function (so ) for a particle must satisfy the linear differential equation (here )
where are constants, and is pure imaginary. The equation is easy to solve: any minimum uncertainly one-dimensional wave function is a Gaussian wave packet, having expectation value of momentum , centered at and having width . ( is defined for a state by .)
That is to say, the minimum uncertainly solution is:
with C the normalization constant.
In fact, the simple harmonic oscillator ground state is just such a minimum uncertainty state, with
Furthermore, it is easy to see that the displaced ground state , with , and writing the normalization constant , must also be a minimum uncertainty state, with the same . (It satisfies the necessary differential equation.) Of course, in contrast to the ground state, this displaced state is no longer an eigenstate of the Hamiltonian, and will therefore change with time.
(Both these states, and , have the same spread in x-space , and the same spread in p-space, the only difference in the p direction being a phase factor for the displaced state.)
What about the higher eigenstates of the oscillator Hamiltonian? They are not minimally uncertain states—for the nth state, , as is easily checked using . So, if we construct a minimally uncertain higher energy state, it will not be an eigenstate of the Hamiltonian.
Exercise: prove for the nth energy eigenstate. (Hint: use creation and annihilation operators.)
Notation: We’ll write
We restrict our attention here to those minimum uncertainty states having the same spatial width as the oscillator ground state—these are what we need, and these are the ones we’ll show to be eigenstates of the annihilation operator. (Actually, more general minimum uncertainty states, known as squeezed states, are also interesting and important, but we’ll not consider them here.)
Suppose that at t = 0 the oscillator wave function is the minimum uncertainty state
centered at in phase space (as defined above for the classical oscillator), and with to give it the same spatial extent as the ground state.
From the preceding section, this satisfies the minimum uncertainty equation
Rearranging this equation (and multiplying by –i) shows it in a different light:
This is an eigenvalue equation! The wave packet is an eigenstate of the operator with eigenvalue . It is not, of course, an eigenstates of either or taken individually.
Furthermore, the operator is just a constant times the annihilation operator —recall
Therefore, this minimally uncertain initial wave packet is an eigenstate of the annihilation operator , with eigenvalue . (By the way, it’s ok for to have complex eigenvalues, because is not a Hermitian operator.)
We can now make the connection with the complex plane representation of the classical operator: the eigenvalue is precisely the parameter labeling the position of the classical operator in phase space in natural dimensionless units!
That is to say, a minimum uncertainty oscillator wave packet
centered at in phase space and having the same spatial extent as the ground state, is an eigenstate of the annihilation operator
with eigenvalue the position of its center in phase space, that is,
Turning now to the time development of the state, it is convenient to use the ket notation
with denoting a minimum uncertainly wave packet (with the same spatial width as the ground state) having those expectation values of position and momentum.
The time development of the ket, as usual, is given by
We shall show that remains an eigenstate of the annihilation operator for all times t: it therefore continues to be a minimum uncertainty wave packet! (And, of course, with constant spatial extent.)
The key point in establishing this is that the annihilation operator itself has a simple time development in the Heisenberg representation,
To prove this, consider the matrix elements of between any two eigenstates of the Hamiltonian
Since the only nonzero matrix elements of the annihilation operator are for , and the energy eigenstates form a complete set, this simple time dependence is true as an operator equation
It is now easy to prove that
is always an eigenstate of :
Therefore the annihilation operator, which at t = 0 had the eigenvalue
corresponding to a minimal wave packet centered at in phase space, evolves in time t to another minimal packet (because it’s still an eigenstate of the annihilation operator), and writing
the new eigenvalue of
Therefore, the center of the wave packet in phase space follows the classical path in time. This is made explicit by equating real and imaginary parts:
So we’ve found Schrödinger’s “best possible” quantum description of a classical oscillator.
We have chosen to work with the original position and momentum variables, and the complex parameter expressed as a function of those variables, throughout. We could have used the dimensionless variables introduced in the lecture on the simple harmonic oscillator,
, This would of course also give a more compact representation, but one more thing to remember.
It’s also common to denote the eigenstates of by very elegant, but we’ve used z to keep reminding ourselves that this eigenvalue, unlike most of those encountered in quantum mechanics, is a complex number. Finally, some use the dimensionless variables differing from The eigenvalue equation for the annihilation operator is very neat in this notation: We’ve avoided it, though, because our recommended textbook, Shankar, uses X, P for the ordinary position and momentum operators.
It’s worth repeating the exercise for the simple case of the oscillator initially at rest a distance from the center. This gives a neat tie-in with the translation operator (defined below).
Let us then take the initial state to be
where is the ground state wave function—so we’ve moved the packet to the right by.
Now do a
It’s clear from this that the translation operator shifts the wave function a distance to the right.
Since , the translation operator can also be written as, and from this it can be expressed in terms of , since
( being Hermitian) so
Therefore the displaced ground state wave function can be written
for real , since is zero for this initial state (the wave function is real).
In the ket notation, we have established that the minimal uncertainty state centered at x0, and having zero expectation value for the momentum, is
But it’s not exactly obvious that this is an eigenstate of with eigenvalue z0! (As it must be.)
It’s worth seeing how to prove that just from the properties of the operators—but to do that, we need a couple of theorems concerning exponentials of operators given in the Appendix.
First, if the commutator [A,B] commutes with A and B, then This result simplifies the right hand side of the above equation, for
where we have used
This is simpler, but it’s still not obvious that we have an eigenstate of : we need the commutator .
The second theorem we need is: if the commutator of two operators itself
commutes with A and B, then
(This is easily proved by expanding the exponential—see the Appendix.)
Applying this to our case,
It follows immediately that is indeed an eigenstate of with eigenvalue . (It must also be correctly normalized because the translation is a unitary operation for real z0.)
How do we generalize this translation operator to an arbitrary state, with nonzero ? Thinking in terms of the complex parameter space z, we need to be able to move in both the x and the p directions, using both and . This is slightly tricky since these operators do not commute, but their commutator is just a number, so (using the theorem proved in the Appendix) this will only affect the overall normalization.
Furthermore, both and are combinations of , so for the generalization of from real to complex z to be unitary, it must have an antihermitian combination of in the exponent—a unitary operator has the form , where H is Hermitian, so iH is antihermitian.
We are led to the conclusion that
conveniently labeling the coherent state using the complex parameter z of its center in phase space. Since this generalized translation operator is unitary, the new state is automatically correctly normalized.
The equation above suggests the possibility of representing the displaced state in the standard energy basis . We can simplify with the same trick used for the spatial displacement case in the last section, that is, the theorem where now :
using since .
It is now straightforward to expand the exponential:
and recalling that the normalized energy eigenstates are
Exercise: Check that this state is correctly normalized, and is an eigenstate of .
Now that we have expressed the eigenstate as a sum over the eigenstates of the Hamiltonian, finding its time development in this representation is straightforward.
which can be written
equivalent to the result derived earlier.
In quantum mechanics, any physical variable is represented by a Hermitian operator. The eigenvalues are real, the eigenstates are orthogonal (or can be chosen to be so for degenerate states) and the eigenstates for a complete set, spanning the space, so any vector in the space can be represented in a unique way as a sum over these states.
The operator is not Hermitian. Its eigenvalues are all the numbers in the complex plane. The eigenstates belonging to different eigenvalues are never orthogonal, as is immediately obvious on considering the ground state and a displaced ground state. The overlap does of course decrease rapidly for states far away in phase space.
The state overlap can be computed using :
and we can then switch the operators using the theorem from the Appendix , then since we’re left with
Finally, using, we can construct a unit operator using the ,
where the integral is over the whole complex plane (this x is not, of course, the original position x, recall for the wave function just displaced along the axis ). Therefore, the span the whole space.
Suppose that the commutator of two operators A, B
where c commutes with A and B, usually it’s just a number, for instance 1 or .
That is to say, the commutator of A with is proportional to itself.
That is reminiscent of the simple harmonic oscillator commutation relation which led directly to the ladder of eigenvalues of H separated by . Will there be a similar “ladder” of eigenstates of A in general?
Assuming A (which is a general operator) has an eigenstate with eigenvalue a,
Applying to the eigenstate :
Therefore, unless it is identically zero, is also an eigenstate of A, with eigenvalue We conclude that instead of a ladder of eigenstates, we can apparently generate a whole continuum of eigenstates, since can be set arbitrarily!
To find more operator identities, premultiply by to find:
This identity is only true for operators A, B whose commutator c is a number. (Well, c could be an operator, provided it still commutes with both A and B).
Our next task is to establish the following very handy identity, which is also only true if [A,B] commutes with A and B:
The proof (due to Glauber, given in Messiah) is as follows:
It is easy to check that the solution to this first-order differential equation equal to one at x = 0 is
so taking x = 1 gives the required identity,
It also follows that provided—as always—that [A, B] commutes with A and B.