- #1
spaghetti3451
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Homework Statement
Consider the following potential, which is symmetric about the origin at ##x=0##:
##V(x) =
\begin{cases}
x^{2}+(x+\frac{d}{2}) &\text{for}\ x < -d/2\\
x^{2} &\text{for}\ -d/2 < x < d/2\\
x^{2}-(x-\frac{d}{2}) &\text{for}\ x > d/2
\end{cases}##
Find the ground state energy and wavefunction for this potential.
Homework Equations
The Attempt at a Solution
For ##x < -d/2##, ##V(x) = (x+\frac{1}{2})^{2}+\frac{2d-1}{4} = (x+\frac{1}{2})^{2}##,and
for ##x > d/2##, ##V(x) = (x-\frac{1}{2})^{2}+\frac{2d-1}{4} = (x-\frac{1}{2})^{2}##.
So, the wavefunctions in these two sectors are the shifted-in-position harmonic oscillator wavefunctions.
So, for ##x < -d/2##, ##\psi_{n}(x) \sim \text{exp}\Big(-\frac{m\omega}{2\hbar}(x-\frac{1}{2})^{2}\Big)\ H_{n}\Big(\sqrt{\frac{m\omega}{\hbar}}(x-\frac{1}{2})\Big)##,
for ##-d/2 < x < d/2##, ##\psi_{n}(x) \sim \text{exp}\Big(-\frac{m\omega}{2\hbar}x^{2}\Big)\ H_{n}\Big(\sqrt{\frac{m\omega}{\hbar}}x\Big)##, and
for ##x > d/2##, ##\psi_{n}(x) \sim \text{exp}\Big(-\frac{m\omega}{2\hbar}(x+\frac{1}{2})^{2}\Big)\ H_{n}\Big(\sqrt{\frac{m\omega}{\hbar}}(x+\frac{1}{2})\Big)##.
But, the wavefunctions at the potential kinks at ##x=-d/2## and ##x=d/2## do not match.
Is there some sorcery of the kink in the potential - perhaps the discontinuity at ##V'(-d/2)## and at ##V(d/2)## - that causing this behaviour of the wavefunction?