Phy5645/Particle in Uniform Magnetic Field

(a) In the symmetric gauge, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{x}=-\tfrac{1}{2}By,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{y}=\tfrac{1}{2}Bx,} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{z}=0.\!}

${\begin{aligned}\left[{{\hat {\Pi }}_{x},{\hat {\Pi }}_{y}}\right]&=\left[{\hat {p}}_{x}-{\frac {e}{c}}A_{x},{\hat {p}}_{y}-{\frac {e}{c}}A_{y}\right]=\left[{\hat {p}}_{x}+{\frac {eB}{2c}}{\hat {y}},{\hat {p}}_{y}-{\frac {eB}{2c}}{\hat {x}}\right]\\&=\left(\left[{\hat {p}}_{x},{\hat {p}}_{y}\right]-\left[{\hat {p}}_{x},{\frac {eB}{2c}}{\hat {x}}\right]+\left[{\frac {eB}{2c}}{\hat {y}},{\hat {p}}_{y}\right]-\left[{\frac {eB}{2c}}{\hat {y}},{\frac {eB}{2c}}{\hat {x}}\right]\right)\\&=-{\frac {eB}{2c}}(-i\hbar )+{\frac {eB}{2c}}(i\hbar )=i\hbar {\frac {eB}{c}}\end{aligned}}$

(b) The Hamiltonian for the system is

${\begin{aligned}{\hat {H}}&={\frac {1}{2m}}\left({\hat {\mathbf {p} }}-{\frac {e}{c}}\mathbf {A} \right)^{2}\\&={\frac {{\hat {\Pi }}_{x}^{2}}{2m}}+{\frac {{\hat {\Pi }}_{y}^{2}}{2m}}+{\frac {{\hat {p}}_{z}^{2}}{2m}}.\end{aligned}}$

If we label the first two terms as ${\hat {H}}_{1}={\frac {{\hat {\Pi }}_{x}^{2}}{2m}}+{\frac {{\hat {\Pi }}_{y}^{2}}{2m}}$, and the last one as ${\hat {H}}_{2}={\frac {{\hat {p}}_{z}^{2}}{2m}}$, then we may write the Hamiltonian as ${\hat {H}}={\hat {H}}_{1}+{\hat {H}}_{2}.$ Using the identity,

${\hat {A}}^{2}+{\hat {B}}^{2}=\left({\hat {A}}-i{\hat {B}}\right)\left({\hat {A}}+i{\hat {B}}\right)-i\left[{\hat {A}},{\hat {B}}\right],$

we may rewrite ${\hat {H}}_{1}$ as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}_1=\frac{1}{2m}\left (\hat{\Pi}_x-i\hat{\Pi}_y\right )\left (\hat{\Pi}_x+i\hat{\Pi}_y\right )+\frac{\hbar eB}{2mc}.}

If we now define the operators,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}=\sqrt{\frac{c}{2\hbar eB}}\left (\hat{\Pi}_x+i\hat{\Pi}_y\right )}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}^\dagger=\sqrt{\frac{c}{2\hbar eB}}\left (\hat{\Pi}_x-i\hat{\Pi}_y\right ),}

this becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}_1=\hbar\omega\left (\hat{a}^\dagger\hat{a}+\tfrac{1}{2}\right ),}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega=\frac{eB}{mc}.} This is just the Hamiltonian for a harmonic oscillator. The contribution to the energy from this term is therefore

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_1=\left (n+\tfrac{1}{2}\right )\hbar\omega.}

The remaining part of the Hamiltonian, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}_2,} is just that of a free particle in one dimension, and thus its contribution to the energy is just Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_2=\frac{\hbar^2k_z^2}{2m}.} The total energy is then just

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=\left (n+\tfrac{1}{2}\right )\hbar\omega+\frac{\hbar^{2}k_z^{2}}{2m}.}

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