[ \hat H = \hbar\omega\Big(\hat a^\dagger\hat a + \tfrac12\Big). ] Problem: Show that the condition (\hat a|0\rangle =0) leads to the normalized ground‑state wavefunction
[ V(x)=\begincases -V_0, & |x|<a\[4pt] 0, & |x|>a, \endcases \qquad V_0>0. ] Solution Manual To Quantum Mechanics Concepts And
[ \psi_0(x)=\Big(\fracm\omega\pi\hbar\Big)^1/4 \exp!\Big[-\fracm\omega2\hbar,x^2\Big]. ] [ \hat H = \hbar\omega\Big(\hat a^\dagger\hat a +
Hamiltonian becomes
[ \psi(x,0)=A \exp!\Big[-\fracx^24\sigma^2+ik_0x\Big], ] \endcases \qquad V_0>
[ V(x)=\begincases 0, & 0<x<L\[4pt] \infty, & \textotherwise \endcases ]