Physics Problems & Solutions: Quantum Mechanics

A measurement of energy E will always satisfy which of the following relationships?

A. E ≤ π²ħ²/8ma²

B. E ≥ π²ħ²/2ma²
C. E = π²ħ²/8ma²
D. E = n²π²ħ²/8ma²
E. E = π²ħ²/2ma²

(GR9277 #53)

Solution:

Wave function, $\psi_n=\sqrt{\frac{2}{a}} \sin \frac{n\pi x}{a}$
Time Independent Schrodinger Equation: $-\frac{\hbar^{2}}{2m}\psi''+ V\psi=E\psi$

Inside the well, V(x) = 0, and

$\\\psi=A\sin kx \\\psi'=Ak\cos kx \\\psi''=-Ak^2\sin kx$

$\\-\frac{\hbar^{2}}{2m}(-Ak^2 \sin kx)=EA\sin kx\rightarrow E=\frac{\hbar^{2}k^2}{2m}$

$\\k=\frac{n\pi}{a}\rightarrow E=\frac{n^2\pi^2 \hbar^{2}}{2ma^2} \\\\n=1,2,3, ...\rightarrow E\geqslant \frac{\pi^2 \hbar^{2}}{2ma^2}$

Answer: B

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