Wavefunctions and Integration

Calculus Level 4

A particle in quantum mechanics confined to two dimensions has the wavefunction

ψ ( x , y ) = C x e − x 2 − y 2 . \psi(x,y) = Cxe^{-x^2-y^2}.

Find the constant C C given the normalization condition ∫ ∣ ψ ∣ 2 d x d y = 1 \int |\psi|^2 \:dx\:dy =1 .

2 π \sqrt{\frac{2}{\pi}} 8 π \sqrt{\frac{8}{\pi}} 2 π \sqrt{2\pi} 8 π \frac{8}{\pi}

Matt DeCross by Matt DeCross , 27, USA

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1 solution

Matt DeCross
Apr 25, 2016

The normalization integral is best done in polar coordinates, using integration by parts:

∫ C 2 x 2 e − 2 x 2 − 2 y 2 d A = ∫ C 2 r 3 cos ⁡ 2 θ e − 2 r 2 d r d θ = C 2 π ∫ r 3 e − 2 r 2 d r = C 2 π ∫ r 2 e − 2 r 2 = C 2 π 1 8 . \int C^2 x^2 e^{-2x^2-2y^2} dA = \int C^2 r^3 \cos^2 \theta e^{-2r^2} dr d\theta = C^2 \pi \int r^3 e^{-2r^2} dr = C^2 \pi \int \frac{r}{2} e^{-2r^2} = C^2 \pi \frac{1}{8}.

Therefore, if the integral is equal to one, C = 8 π . C = \sqrt{\frac{8}{\pi}}.

8 Helpful 1 Interesting 2 Brilliant 0 Confused

Admire the gamma function .

Syed Shahabudeen - 3 years, 6 months ago

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