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Given a Gaussian wave function: Y(x) = e-mwx?/h %3D Where a is a positive constant 1) Determine the energy eigenvalues. 2) Determine (p") , where p is the momentum.

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Question

Given a Gaussian wave function:
Y(x) = e-mwx?/h
%3D
Where a is a positive constant
1) Determine the energy eigenvalues.
2) Determine (p") , where p is the momentum.

Expert Solution

Step 1

1) The wavefunction is given as :
$ψ (x) = e^{- \frac{m ω x^{2}}{h}}$

The energy eigenvalue is calculated as :

$<E> = <ψ (x) | \hat{E} | ψ (x)> <E> = A^{2} \int_{- \infty}^{\infty} e^{- \frac{m ω x^{2}}{h}} (- \frac{h^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}) e^{- \frac{m ω x^{2}}{h}} d x <E> = - \frac{h^{2}}{2 m} A^{2} \int_{- \infty}^{\infty} e^{- \frac{m ω x^{2}}{h}} (\frac{\partial^{2}}{\partial x^{2}}) e^{- \frac{m ω x^{2}}{h}} d x <E> = - \frac{h^{2}}{2 m} A^{2} \int_{- \infty}^{\infty} e^{- \frac{m ω x^{2}}{h}} (- \frac{2 m ω}{h}) \frac{\partial}{\partial x} (x e^{- \frac{m ω x^{2}}{h}}) d x <E> = \frac{h^{2}}{2 m} \frac{2 m ω}{h} A^{2} \int_{- \infty}^{\infty} e^{- \frac{m ω x^{2}}{h}} [e^{- \frac{m ω x^{2}}{h}} - \frac{2 m ω x^{2}}{h} e^{- \frac{m ω x^{2}}{h}}] d x <E> = h ω A^{2} \int_{- \infty}^{\infty} [e^{- \frac{2 m ω x^{2}}{h}} - \frac{2 m ω x^{2}}{h} e^{- \frac{2 m ω x^{2}}{h}}] d x$

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