Example 7.1 Changed frequency
\(V(x)=2m\omega^2x^2\). Find the energies and wave functions.
Solution
\(V=\frac12m(2\omega)^2x^2\) \(\Rightarrow\) frequency \(2\omega\): \(E_n=\big(n+\tfrac12\big)2\hbar\omega\).
\(\phi_n=\Big(\dfrac{2m\omega}{\pi\hbar}\Big)^{1/4}\dfrac{1}{\sqrt{2^nn!}}H_n(\alpha x)\exp\Big(-\dfrac{m\omega x^2}{\hbar}\Big)\), \(\alpha=\sqrt{\dfrac{2m\omega}{\hbar}}\).
Example 7.2 Energy from the wave function (1)
\(\phi(x)\propto x\exp\Big(-\dfrac{m\omega x^2}{\hbar}\Big)\). Find its energy.
Solution
Example 7.3 Energy from the wave function (2)
\(\psi(x)\propto x^2\exp\Big(-\dfrac{m\omega x^2}{\hbar}\Big)\). What energies can be measured?
Solution
Frequency \(2\omega\); \(x^2=\dfrac{H_2+2H_0}{4}\) \(\Rightarrow\psi\propto c_2\left|2\right\rangle+c_0\left|0\right\rangle\) — not a single eigenstate.
Outcomes: \(E_2=\tfrac52\cdot2\hbar\omega=5\hbar\omega\) or \(E_0=\tfrac12\cdot2\hbar\omega=\hbar\omega\).
Example 7.4 Measurement on a superposition
\(\left|\psi\right\rangle=3\left|0\right\rangle+4\left|1\right\rangle\) in \(V=\frac12m\omega^2x^2\). Outcomes and probabilities of energy?
Solution
Example 7.5 Half oscillator
\(V=\infty\) for \(x\le0\); \(\frac12m\omega^2x^2\) for \(x>0\). Find the ground and first excited energies.
Solution
Allowed solutions must vanish at \(x=0\): only the odd oscillator states, \(n=1,3,5,\ldots\)
Ground: \(E=\tfrac32\hbar\omega\); first excited: \(E=\tfrac72\hbar\omega\).
Example 7.6 Average energy in a half oscillator
\(V=\infty\) (\(x\le0\)), \(2m\omega^2x^2\) (\(x>0\)); \(\left|\psi\right\rangle=3\left|g\right\rangle+4\left|f\right\rangle\) (ground, first excited). Find \(\left\langle E\right\rangle\).
Solution
Frequency \(2\omega\); \(\left|g\right\rangle=\left|1\right\rangle\): \(E=\tfrac32\cdot2\hbar\omega=3\hbar\omega\); \(\left|f\right\rangle=\left|3\right\rangle\): \(E=\tfrac72\cdot2\hbar\omega=7\hbar\omega\).
\(\left\langle E\right\rangle=\dfrac{9\cdot3+16\cdot7}{25}\hbar\omega=\dfrac{139}{25}\hbar\omega\).
Example 7.7 Time evolution
At \(t=0\), \(\left|\psi\right\rangle=3\left|1\right\rangle+4\left|2\right\rangle\). Find the state at \(t=\pi/\omega\).
Solution
\(\left|\psi(t)\right\rangle=3e^{-3i\omega t/2}\left|1\right\rangle+4e^{-5i\omega t/2}\left|2\right\rangle\).
At \(\omega t=\pi\): \(e^{-3i\pi/2}=i\), \(e^{-5i\pi/2}=-i\) \(\Rightarrow\left|\psi\right\rangle=3i\left|1\right\rangle-4i\left|2\right\rangle\).
Example 7.8 Charged oscillator in an electric field
A charge \(q\) on a spring (\(k\), \(m\)) is placed in a uniform field \(E_0\) (\(V=0\) at \(x=0\)). (a) Write \(H\). (b) Find the energies.
Solution
(a) \(L=\tfrac12m\dot x^2-\tfrac12kx^2+qE_0x\) \(\Rightarrow H=\dfrac{p^2}{2m}+\dfrac12m\omega^2x^2-qE_0x\).
(b) Complete the square: \[H=\frac{p^2}{2m}+\frac12m\omega^2\Big(x-\frac{qE_0}{m\omega^2}\Big)^2-\frac{q^2E_0^2}{2m\omega^2}.\]
Same frequency, shifted centre: \(E_n=\big(n+\tfrac12\big)\hbar\omega-\dfrac{q^2E_0^2}{2m\omega^2}\).
Example 7.9 \(\left\langle x\right\rangle\) and \(\left\langle x^2\right\rangle\) in \(\left|n\right\rangle\)
Solution
\(\left\langle x\right\rangle=\sqrt{\dfrac{\hbar}{2m\omega}}\big(\sqrt n\left\langle n\middle|n-1\right\rangle +\sqrt{n+1}\left\langle n\middle|n+1\right\rangle \big)=0\).
\(x^2=\dfrac{\hbar}{2m\omega}\big(a^2+a^{\dagger2}+aa^\dagger+a^\dagger a\big)=\dfrac{\hbar}{2m\omega}\big(a^2+a^{\dagger2}+2N+1\big)\).
\(\left\langle x^2\right\rangle=\dfrac{\hbar}{2m\omega}(2n+1)\) (only the \(2N+1\) term survives).
Example 7.10 \(\left\langle x\right\rangle\), \(\left\langle x^2\right\rangle\) in a superposition (1)
\(\left|\psi\right\rangle=\sqrt{\tfrac23}\left|1\right\rangle+\sqrt{\tfrac13}\left|2\right\rangle\). Find \(\left\langle x\right\rangle\), \(\left\langle x^2\right\rangle\).
Solution
\(\left\langle a\right\rangle=\left\langle a^\dagger\right\rangle=\sqrt{\tfrac23}\sqrt{\tfrac13}\sqrt2=\tfrac23\) \(\Rightarrow\left\langle x\right\rangle=\sqrt{\dfrac{\hbar}{2m\omega}}\cdot\dfrac43\).
\(\left\langle a^2\right\rangle=0\) (no states differing by 2);
