Phy5645/Particle in Uniform Magnetic Field

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An electron moves in magnetic field which is in the z direction, ${\overrightarrow {B}}=B{\hat {z}}$, and the Landau gauge ${\overrightarrow {A}}=({\frac {-By}{2}},{\frac {Bx}{2}},0)$

• Evaluate $\left[{\Pi _{x},\Pi _{y}}\right]$
• Using the hamiltonian and commutation relation obtained in a), obtain the energy eigenvalues.

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• According to the Ladau gauge, ${\text{A}}_{x}={\frac {-By}{2}}{\text{ A}}_{y}={\frac {Bx}{2}}{\text{ A}}_{z}=0$

$\left[{\Pi _{x},\Pi _{y}}\right]=\left[{\left({P_{x}-{\frac {e}{c}}A_{x}}\right),\left({P_{y}-{\frac {e}{c}}A_{y}}\right)}\right]$

$=\left[{\left({P_{x}+{\frac {eBy}{2c}}}\right),\left({P_{y}-{\frac {eBx}{2c}}}\right)}\right]$

$=\left\lbrace {\left[{P_{x},P_{y}}\right]-\left[{P_{x},{\frac {eBx}{2c}}}\right]+\left[{{\frac {eBy}{2c}},P_{y}}\right]-\left[{{\frac {eBy}{2c}},{\frac {eBx}{2c}}}\right]}\right\rbrace$

${\text{= -}}{\frac {eB}{2c}}(-i\hbar )+{\frac {eB}{2c}}(i\hbar )$ ${\text{=}}i\hbar {\frac {eB}{c}}$

• The hamiltonian fot the system is;

$H={\frac {({\overrightarrow {P}}-{\frac {e{\overrightarrow {A}}}{c}})^{2}}{2m}}$

$={\frac {\left({P_{x}-{\frac {e}{c}}A_{x}}\right)^{2}}{2m}}+{\frac {\left({P_{y}-{\frac {e}{c}}A_{y}}\right)^{2}}{2m}}+{\frac {\left({P_{z}-{\frac {e}{c}}A_{z}}\right)^{2}}{2m}}$

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{=}\frac{\Pi _{x}^{2}}{2m}+\frac{\Pi _{y}^{2}}{2m}+\frac{P_{z}^{2}}{2m}}

If we define 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 \text{H}_{2}=\frac{P_{z}^{2}}{2m}} , The hamiltonian will be 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=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}}

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