String Waves and Matter Waves — Question 6

Question 6

Classical mode quantization does not quantize the amplitude-dependent string energy

A classical string mode is $y=A\sin(n\pi x/L)\cos\omega_nt$, with tension $\mathcal T$, linear density $\mu$ and $\omega_n^2=(\mathcal T/\mu)(n\pi/L)^2$. At maximum displacement its energy is $U=(\mathcal T/2)\int_0^L(\partial_xy)^2dx$.

1. Evaluate the mode energy as a function of amplitude.

2. Compare amplitude doubling for the string with multiplying a normalized particle eigenfunction by two.

Question 6 - Solution

Classical mode quantization does not quantize the amplitude-dependent string energy

(a) Method and calculation

$$U=\frac{\mathcal T A^2n^2\pi^2}{2L^2}\int_0^L\cos^2(n\pi x/L)dx.$$$$\boxed{U=\frac{\mathcal T A^2n^2\pi^2}{4L}=\frac14\mu L\omega_n^2A^2}.$$

(b) Interpretation and further calculation

$$A\to2A\Rightarrow\boxed{U\to4U}\quad\text{for the classical string}.$$$$\psi\to2\psi:\ \|2\psi\|^2=4\Rightarrow\frac{2\psi}{\|2\psi\|}=\psi\Rightarrow\boxed{E_n\text{ unchanged}}.$$

Physical check

Classical standing-wave frequencies are discrete, but their classical energies vary continuously with amplitude. A quantum state must be normalized before interpreting it.