Answer to Question #236631 in Quantum Mechanics for dumela

Question #236631

Consider the function πœ“(πœƒ) of the angular variable πœƒ, restricted to the interval βˆ’πœ‹ ≀ πœƒ ≀ πœ‹. If the wave functions satisfy the condition πœ“(πœ‹) = πœ“(βˆ’πœ‹), show that the operator 𝐿 = ℏ / 𝑖. 𝑑 / π‘‘πœƒ has a real expectation value.


1
Expert's answer
2021-09-14T09:36:08-0400

The equation for eigenfunctions is given by

Lψ=λψL\psi=\lambda\psi

ℏiddθψ=λψ\frac{\hbar}{i}\frac{d}{d\theta}\psi=\lambda\psi

ψ(ΞΈ)=Cei/ℏλθ\psi(\theta)=Ce^{i/\hbar\lambda\theta}

ψ(Ο€)=Cei/ℏλπ,ψ(βˆ’Ο€)=Ceβˆ’i/ℏλπ\psi(\pi)=Ce^{i/\hbar\lambda \pi},\quad \psi(-\pi)=Ce^{-i/\hbar\lambda \pi}

ei/ℏλπ=eβˆ’i/ℏλπe^{i/\hbar\lambda \pi}=e^{-i/\hbar\lambda \pi}

e2i/ℏλπ=1e^{2i/\hbar\lambda \pi}=1

2/ℏλπ=2Ο€n2/\hbar\lambda \pi=2\pi n

The expectation values of operator

Ξ»=ℏn=real\lambda=\hbar n=\rm real


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