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 [{\Pi _{x},\Pi _{y}} \right ]&=\left [p_{x}-\frac{e}{c}A_{x},P_{y}-\frac{e}{c}A_{y}\right ]=\left [p_{x}+\frac{eBy}{2c},p_{y}-\frac{eBx}{2c}\right ] \\ &=\left (\left [p_{x},p_{y}\right ]-\left [p_{x},\frac{eBx}{2c}\right ]+\left [\frac{eBy}{2c},p_{y}\right ]-\left [\frac{eBy}{2c},\frac{eBx}{2c}\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} H&=\frac{1}{2m}\left (\mathbf{P}-\frac{e}{c}\mathbf{A}\right )^2 \\ &=\frac{\Pi_{x}^{2}}{2m}+\frac{\Pi_{y}^{2}}{2m}+\frac{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 \text{H}_{1}=\frac{\Pi_{x}^{2}}{2m}+\frac{\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 H_{2}=\frac{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 H=H_{1}+H_{2}\!} .

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 H_{1}=\frac{\Pi_{x}^{2}}{2m}+\frac{\Pi_{y}^{2}}{2m}}

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 =\frac{1}{2m}\left ({\frac{c\Pi _{x}}{eB}} \right )^{2}\left ({\frac{e^{2}B^{2}}{c^{2}}} \right )+\frac{\Pi _{y}^{2}}{2m}}

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 =\frac{\Pi _{y}^{2}}{2m}+\frac{1}{2m}\left ({\frac{m^{2}}{m^{2}}} \right )\left ({\frac{e^{2}B^{2}}{c^{2}}} \right )\left ({\frac{c\Pi _{x}}{eB}} \right )^{2}}

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 =\frac{\Pi _{y}^{2}}{2m}+\frac{1}{2}m\left ({\frac{eB}{cm}} \right )^{2}\left ({\frac{c\Pi _{x}}{eB}} \right )^{2}}

Then the Hamiltonian will look like 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 \text{H}_{1}=\frac{\Pi _{y}^{2}}{2m}+\frac{1}{2}m \tilde{w^{2}} \tilde{x^{2}}} 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 \tilde{w}= \left ({\frac{eB}{cm}} \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 \tilde{x}= \left ({\frac{c\Pi _{x}}{eB}} \right )} .

As we know, 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 \text{H}\Psi =E\Psi \!}

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 \text{H=}\hbar \left ({\frac{eB}{cm}} \right )(n+\frac{1}{2}) + \frac{P_{z}^{2}}{2m} }

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 \text{H=}\hbar \left ({\frac{eB}{cm}} \right )(n+\frac{1}{2})+\frac{\hbar ^{2}k^{2}}{2m}}

So now we can write that;

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 \text{E}_{k,n}=\hbar\frac{eB}{cm}(n+\frac{1}{2})+\frac{\hbar ^{2}k^{2}}{2m}}

Back to Charged Particles in an Electromagnetic Field.