Physics Problems & Solutions: Quantum Mechanics - Normalization of Wavefunction

Eigenfunction for a rigid dumbbell rotating about its center have a φ dependence of the form ψ(φ) = Ae^imφ, where m is a quantum number and A is a constant. Which of the following values of A will properly normalize the eigenfunction?

A. √(2π)
B. 2π
C. (2π)²
D. 1/√(2π)
E. 1/(2π)

(GR8677 #28)

Solution:
Normalized Wavefunction: $\int |\psi(x)|^2dx=\int \psi(x)\psi^*(x)dx=1$
$\\\psi(\phi)=Ae^{im\phi} \\\psi^*(\phi)=Ae^{-im\phi}$

$\\\int_{0}^{2\pi} |\psi(\phi)|^2d\phi=\int_{0}^{2\pi} |A|^2d\phi=1 \\|A|^2\int_{0}^{2\pi}d\phi=|A|^2 2\pi=1\rightarrow A = \frac{1}{\sqrt{2\pi}}$

Answer: D

Quantum Mechanics - Normalization of Wavefunction

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