Phy5645/Particle in Uniform Magnetic Field

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

  • Evaluate 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 \left [{\Pi _{x},\Pi _{y}} \right ]}
  • Using the hamiltonian and commutation relation obtained in a), obtain the energy eigenvalues.
  • According to the Ladau 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 \text{A}_{x}=\frac{-By}{2}\text{ A}_{y}=\frac{Bx}{2}\text{ 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 \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 ]}

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

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 =\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 }

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


  • The hamiltonian fot 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 H=\frac{(\overrightarrow{P}-\frac{e\overrightarrow{A}}{c})^{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{\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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