The wave function of a particle is confined to a harmonic oscillator potential given by \[ v=A e^{\left(\frac{-m x^{2}}{2 \hbar}-\frac{i w t}{2}\right)} \] (i) Verify that this is a solution to the Schrodinger equation \[ -\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} \psi(x, t)+V(x) \psi(x, t)=E \psi(x, t) \] (ii) Normalize the wave function \( \psi(x)=A x\left(l-\frac{x}{2}\right) \) for \( 0 \leq x \leq \frac{l}{2} \)
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The wave function for a quantum particle confined to moving in a one-dimensional box is $$ \psi(x)=A \sin \left(\frac{n \pi x}{L}\right) $$ Use the normalization condition on $\psi$ to show that $$ A=\sqrt{\frac{2}{L}} $$ Suggestion: Because the length of the box is $L,$ the wave function is zero for $x<0$ and for $x>L$ , so the normalization condition (Eq. 41.7$)$ reduces to $$ \int_{0}^{L}|\psi|^{2} d x=1 $$
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