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.\!}
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 \begin{align} \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{align} }
(b) The Hamiltonian for the system is
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 \begin{align} \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{align} }
If we label the first two terms 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{\hat{\Pi}_{x}^{2}}{2m}+\frac{\hat{\Pi}_{y}^{2}}{2m}} , and the last one 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}_{2}=\frac{\hat{p}_{z}^{2}}{2m}} , then we may write the Hamiltonian 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}=\hat{H}_{1}+\hat{H}_{2}.} Using the identity,
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}^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 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} 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},}
which is just the result that we obtained working in the Landau gauge, as expected.