Principles of Quantum Mechanics · Problem 10.1.3

Systems with N Degrees of Freedom — R. Shankar

Physics — Quantum Mechanics · coupled-oscillators-normal-coordinates

Problem

Consider the Hamiltonian of the coupled mass system:

\mathscr{H}=\frac{p_1^2}{2m}+\frac{p_2^2}{2m}+\frac{1}{2}m\omega^2[x_1^2+x_2^2+(x_1-x_2)^2]

We know from Example 1.8.6 that

\mathscr{H}

can be decoupled if we use normal coordinates

x_{\mathrm{I,II}}=\frac{x_1\pm x_2}{2^{1/2}}

and the corresponding momenta

p_{\mathrm{I,II}}=\frac{p_1\pm p_2}{2^{1/2}}

(1) Rewrite

\mathscr{H}

in terms of normal coordinates. Verify that the normal coordinates are also canonical, i.e., that

\{x_i,p_j\}=\delta_{ij}\text{ etc.};\quad i,j=\mathrm{I,II}

Now quantize the system, promoting these variables to operators obeying

[X_i,P_j]=i\hbar\delta_{ij}\text{ etc.};\quad i,j=\mathrm{I,II}

Write the eigenvalue equation for

H

in the simultaneous eigenbasis of

X_\mathrm{I}

and

X_\mathrm{II}

. (2) Quantize the system directly, by promoting

x_1

x_2

p_1

, and

p_2

to quantum operators. Write the eigenvalue equation for

H

in the simultaneous eigenbasis of

X_1

and

X_2

. Now change from

x_1

x_2

(and of course

\partial/\partial x_1

\partial/\partial x_2

) to

x_\mathrm{I}

x_\mathrm{II}

(and

\partial/\partial x_\mathrm{I}

\partial/\partial x_\mathrm{II}

) in the differential equation. You should end up with the result from part (1).