[ Π x , Π y ] = [ ( P x − e c A x ) , ( P y − e c A y ) ] {\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]}
= [ ( P x + e B y 2 c ) , ( P y − e B x 2 c ) ] {\displaystyle =\left[{\left({P_{x}+{\frac {eBy}{2c}}}\right),\left({P_{y}-{\frac {eBx}{2c}}}\right)}\right]}
= { [ P x , P y ] − [ P x , e B x 2 c ] + [ e B y 2 c , P y ] − [ e B y 2 c , e B x 2 c ] } {\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 }
= - e B 2 c ( − i ℏ ) + e B 2 c ( i ℏ ) {\displaystyle {\text{= -}}{\frac {eB}{2c}}(-i\hbar )+{\frac {eB}{2c}}(i\hbar )} = i ℏ e B c {\displaystyle {\text{=}}i\hbar {\frac {eB}{c}}}
H = ( P → − e A → c ) 2 2 m {\displaystyle H={\frac {({\overrightarrow {P}}-{\frac {e{\overrightarrow {A}}}{c}})^{2}}{2m}}}
= ( P x − e c A x ) 2 2 m + ( P y − e c A y ) 2 2 m + ( P z − e c A z ) 2 2 m {\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}}}
= Π x 2 2 m + Π y 2 2 m + P z 2 2 m {\displaystyle {\text{=}}{\frac {\Pi _{x}^{2}}{2m}}+{\frac {\Pi _{y}^{2}}{2m}}+{\frac {P_{z}^{2}}{2m}}}
If we define first two terms as H 1 = Π x 2 2 m + Π y 2 2 m {\displaystyle {\text{H}}_{1}={\frac {\Pi _{x}^{2}}{2m}}+{\frac {\Pi _{y}^{2}}{2m}}} , and the last one as H 2 = P z 2 2 m {\displaystyle {\text{H}}_{2}={\frac {P_{z}^{2}}{2m}}} , The Hamiltonian will be H=H 1 + H 2 {\displaystyle {\text{H=H}}_{1}+H_{2}\!} .
H 1 = Π x 2 2 m + Π y 2 2 m {\displaystyle H_{1}={\frac {\Pi _{x}^{2}}{2m}}+{\frac {\Pi _{y}^{2}}{2m}}}
= 1 2 m ( c Π x e B ) 2 ( e 2 B 2 c 2 ) + Π y 2 2 m {\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}}}
= Π y 2 2 m + 1 2 m ( m 2 m 2 ) ( e 2 B 2 c 2 ) ( c Π x e B ) 2 {\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}}
= Π y 2 2 m + 1 2 m ( e B c m ) 2 ( c Π x e B ) 2 {\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 H 1 = Π y 2 2 m + 1 2 m w 2 ~ x 2 ~ {\displaystyle {\text{H}}_{1}={\frac {\Pi _{y}^{2}}{2m}}+{\frac {1}{2}}m{\tilde {w^{2}}}{\tilde {x^{2}}}} where w ~ = ( e B c m ) {\displaystyle {\tilde {w}}=\left({\frac {eB}{cm}}\right)} and x ~ = ( c Π x e B ) {\displaystyle {\tilde {x}}=\left({\frac {c\Pi _{x}}{eB}}\right)} .
As we know, H Ψ = E Ψ {\displaystyle {\text{H}}\Psi =E\Psi \!}
H= ℏ ( e B c m ) ( n + 1 2 ) + P z 2 2 m {\displaystyle {\text{H=}}\hbar \left({\frac {eB}{cm}}\right)(n+{\frac {1}{2}})+{\frac {P_{z}^{2}}{2m}}}
H= ℏ ( e B c m ) ( n + 1 2 ) + ℏ 2 k 2 2 m {\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;
E k , n = ℏ e B c m ( n + 1 2 ) + ℏ 2 k 2 2 m {\displaystyle {\text{E}}_{k,n}=\hbar {\frac {eB}{cm}}(n+{\frac {1}{2}})+{\frac {\hbar ^{2}k^{2}}{2m}}}
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06dd4d8fafc813d02f47222b660db6df9f137767)
![{\displaystyle =\left[{\left({P_{x}+{\frac {eBy}{2c}}}\right),\left({P_{y}-{\frac {eBx}{2c}}}\right)}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a34c905ac1b455bdea93e599eec38942105b8b0e)
![{\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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/794751e57b19114341fa699ad17bf4a21d7e8414)
, and the last one as 
,
The Hamiltonian will be 
.
where 
and 
.
