Conjugate Variables and Fourier Transforms in Classical Physics

Let q be a generalized coordinate with a conjugate momentum p and a potential resulting in a periodic motion of q. What is the meaning of the Fourier transform of q(t) over its period? Can this be interpreted as the distribution of momenta?

I know this is the case in QM but what happens in the Classical case? For instance for a particle in a box (classical) with an initial velocity of p/m, the motion is a triangle wave. The Fourier transform is a squared sinc function. As the triangle wave gets smaller, the F transform spreads out and vice versa. Very nice but the only two momenta for the particle are p and -p as the particle either moves up with constant speed or moved down at constant speed.

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Why do you say that this is the case in QM? Very loosely speaking, the momentum is the Fourier transform of $q$, not of $V$. – yohBS Jan 2 '12 at 19:25
He apparently means Fourier transform over the time coordinate. – Luboš Motl Jan 2 '12 at 19:49
He is confusing Fourier transform of the wavefunction in q with Fourier transform of q in time. The time Fourier transform of q is the semiclassical matrix element of q between nearby large N energy levels in the correspondence limit, as in Heisenberg's Matrix Mechanics. – Ron Maimon Jan 3 '12 at 8:27

The Fourier transform of the wavefunction as a function of q is the wavefunction as a functon of x. The Fourier transform of q(t) is a different thing altogether, and it quantum mechanically corresponds to the matrix elements of the q operator between adjacent energy states.

When the motion is periodic, the Fourier transform over all time is a series of delta function spikes at the locations $2\pi n\over T$ where n is an integer, and T is the classical period. The Fourier series coefficients C_n are the coefficients of these delta-spikes.

The quantum mechanical analogs of $C_n$ for the motion with period T are the matrix elements of the q operator near the diagonal in the energy representation, for the value of N which corresponds to the classical motion

$$q_{N,N+n} = C_n$$

valid in the limit where $n<<N$. The value of N is J/h where J is the action of the classical orbit.

This correspondence was Heisenberg's central idea in 1925, and he built modern quantum mechanics around this. You can derive it from modern quantum mechanics easily, although this is historically backwards.

If you have a q(t), this must be a wavepacket consisting of a superpositon of a gazillion nearby energy levels close to the energy level corresponding to the classical orbit energy E. From the Heisenberg equation of motion, the matrix elements of q(t) are periodic with periods E_n-E, which is $(n-m)2\pi/T$ by the correspondence principle.

If the state $|\psi\rangle$ is a superposition of many energy levels:

$$|\psi\rangle = \sum_k C_k |E_k>$$

With all relevant $E_i$ only infinitesimally different from $E_N = E$, If you look at the expected value of q as a function of time in this smeared out state, you get

$$\langle \psi | q(t) |\psi \rangle = \sum_k C_n^* C_{n-k} \langle N| q |N-k\rangle e^{i(E_N - E_{N-k})t}$$

Which, using the correspondence rule for the energy level spacings at large quantum numbers $(E_N - E_{N-k}) = {2\pi k \over T}$, gives the classical Fourier series coefficients of q(t):

$$q(t) = \sum_n e^{i {2\pi n\over T }} \langle N|q_|N-k\rangle \sum_k C_k^* C_{k-n}$$

When the C's are slowly varying functions of n, so that the superposition does not involve macroscopically separate positions, the sum on the inside is essentially constant, it is independent of n, so that the Fourier series coefficients are identifiable as the matrix elements of q.

In words, smearing the wavefunction over many nearby energy levels tells you that the off-diagonal periodic quantity has a Fourier coefficient proportional to the matrix element in the semiclassical approximation. The natural deductive path historically went the other way, this was how Heisenberg interpreted the matrix elements in the energy basis, as generalizations of Fourier coefficients of classical quantities in time.

So, in particular, the Fourier transform of the saw-tooth form gives you the matrix elements for $\langle E_n|q|E_m\rangle$ for the square-well energy levels for large quantum numbers.

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