CritPt / Challenge_26_main / Consider an electron (charge −e and mass m) in a perfectly spherical…

# Problem setup:
Consider an electron (charge −e and mass m) in a perfectly spherical conducting cavity of radius R. The electron is subject to a uniform magnetic field B=B(z)^ as well as a quadrupole electric potential, which confines it to the center of the cavity. Assume that the quadrupole potential is weak so that the Hamiltonian for the electron in the absence of the cavity is (in natural units)
\begin{equation}
    H = \omega_+ \left(a^\dagger a + \frac{1}{2}\right) + \omega_z \left(a_z^\dagger a_+ + \frac{1}{2}\right) - \omega_- \left(b^\dagger b + \frac{1}{2}\right) + \frac{geB}{2m}\frac{\sigma_z}{2},
\end{equation}
where a^(†) is the creation operator for cyclotron motion, a_(z)^(†) is the creation operator for axial motion along the magnetic field direction, b^(†) is the creation operator for magnetron motion, g is the g-factor of the electron, and σ_(z) is a Pauli matrix. We assume ω_(−)≪ω_(z)≪ω_(+) and that ω_(+)≈ω_(c)^((0))=eB/m, the classical cyclotron frequency. The magnetron motion is only metastable, but the lifetime is extremely long and the frequency is very small, so we refer to the ''ground state'' |0⟩ as the spin-down state with zero cyclotron, axial, and magnetron number, i.e., a|0⟩=a_(z)|0⟩=b|0⟩=0.

The interaction of the electron with the quantized radiation field, which is subject to spherical conducting boundary conditions, will shift the cyclotron frequency of the electron. The \emph{cavity shift} is defined as
\begin{equation}
    \Delta \omega_c \equiv {\rm Re} [ \Delta E_1 - \Delta E_0],
\end{equation}
where ΔE₁ is the shift in the energy of the first excited cyclotron state a^(†)|0⟩ due to the presence of the cavity (which may be complex because the cavity can affect the lifetime of the excited state), and ΔE₀ is the shift in the energy of the ground state |0⟩. The cavity shift is very small, but it is a necessary ingredient in accurately extracting the magnetic moment of the electron from measurements of quantum transitions of a single trapped electron, given the incredible precision of the experiments.

By quantizing the radiation field and using non-relativistic perturbation theory, the cavity shift may be computed quantum mechanically. The calculation is facilitated by the use of the dipole approximation, where we assume the electron is confined at the exact center of the cavity in all three dimensions; the long-distance approximation (1)/(mR)≪1, which states that the electron's Compton wavelength is much smaller than the size of the cavity; and the weak-field approximation (eB)/(m²)≪1.

Assume ω_(z)/ω_(c)^((0))=10^(−4), ω_(−)/ω_(c)^((0))=5×10^(−9), $R = 1 \ {\rm cm}$ and $B = 5 \ {\rm T}$.

# Main problem:

Compute the dimensionless cavity shift Δω_(c)/ω_(c)^((0)) in non-relativistic quantum mechanics to three significant figures.

# Problem setup:
Consider an electron (charge $-e$ and mass $m$) in a perfectly spherical conducting cavity of radius $R$. The electron is subject to a uniform magnetic field $\mathbf{B} = B\hat{\mathbf{z}}$ as well as a quadrupole electric potential, which confines it to the center of the cavity. Assume that the quadrupole potential is weak so that the Hamiltonian for the electron in the absence of the cavity is (in natural units)
\begin{equation}
    H = \omega_+ \left(a^\dagger a + \frac{1}{2}\right) + \omega_z \left(a_z^\dagger a_+ + \frac{1}{2}\right) - \omega_- \left(b^\dagger b + \frac{1}{2}\right) + \frac{geB}{2m}\frac{\sigma_z}{2},
\end{equation}
where $a^\dagger$ is the creation operator for cyclotron motion, $a_z^\dagger$ is the creation operator for axial motion along the magnetic field direction, $b^\dagger$ is the creation operator for magnetron motion, $g$ is the $g$-factor of the electron, and $\sigma_z$ is a Pauli matrix. We assume $\omega_- \ll \omega_z \ll \omega_+$ and that $\omega_+ \approx \omega_c^{(0)} = eB/m$, the classical cyclotron frequency. The magnetron motion is only metastable, but the lifetime is extremely long and the frequency is very small, so we refer to the ''ground state'' $|0 \rangle$ as the spin-down state with zero cyclotron, axial, and magnetron number, i.e., $a |0 \rangle = a_z |0 \rangle = b |0\rangle = 0$.

The interaction of the electron with the quantized radiation field, which is subject to spherical conducting boundary conditions, will shift the cyclotron frequency of the electron. The \emph{cavity shift} is defined as
\begin{equation}
    \Delta \omega_c \equiv {\rm Re} [ \Delta E_1 - \Delta E_0],
\end{equation}
where $\Delta E_1$ is the shift in the energy of the first excited cyclotron state $a^\dagger |0\rangle$ due to the presence of the cavity (which may be complex because the cavity can affect the lifetime of the excited state), and $\Delta E_0$ is the shift in the energy of the ground state $|0 \rangle$. The cavity shift is very small, but it is a necessary ingredient in accurately extracting the magnetic moment of the electron from measurements of quantum transitions of a single trapped electron, given the incredible precision of the experiments.

By quantizing the radiation field and using non-relativistic perturbation theory, the cavity shift may be computed quantum mechanically. The calculation is facilitated by the use of the dipole approximation, where we assume the electron is confined at the exact center of the cavity in all three dimensions; the long-distance approximation $\frac{1}{mR} \ll 1$, which states that the electron's Compton wavelength is much smaller than the size of the cavity; and the weak-field approximation $\frac{eB}{m^2} \ll 1$.

Assume $\omega_z/\omega_c^{(0)} = 10^{-4}$, $\omega_-/\omega_c^{(0)} = 5 \times 10^{-9}$, $R = 1 \ {\rm cm}$ and $B = 5 \ {\rm T}$.

# Main problem:

Compute the dimensionless cavity shift $\Delta \omega_c/\omega_c^{(0)}$ in non-relativistic quantum mechanics to three significant figures.