凝聚态场论常用公式（8）：Landau能级的Landau规范

2023-03-26 14:30 作者:打电动的阿伟嘻嘻嘻 0人读过 | 我要投稿

考虑二维自由电子气在强磁场下的运动. 我们给系统的对称性限制为：x,z方向的平移对称性.

由对称性限制取Landau规范： $A_x%3D-By%2C%5C%20A_y%3DA_z%3D0.$

代入磁场中的Schrodinger方程

$%5Chat%7BH%7D%5Cpsi%3D%5Cfrac%7B1%7D%7B2m%7D%5B(%5Chat%7Bp%7D_x%2B%5Cfrac%7BeB%7D%7Bc%7Dy)%5E2%2B%5Chat%7Bp%7D%5E2_y%2B%5Chat%7Bp%7D%5E2_z%5D%5Cpsi-%5Cmu%20s_zB%5Cpsi%3DE%5Cpsi%2C$

$s_z%E6%98%AF%E7%94%B5%E5%AD%90%E7%9A%84%E8%87%AA%E6%97%8Bz%E5%88%86%E9%87%8F%2C%20-%5Cmu%20s_zB%E6%98%AF%E7%94%B5%E5%AD%90%E8%87%AA%E6%97%8B%E7%A3%81%E7%9F%A9%E5%9C%A8%E7%A3%81%E5%9C%BAB%E7%9A%84%E8%83%BD%E9%87%8F.$

由x,z方向的平移对称性得：

$%5Cpsi(x%2Cy%2Cz)%3De%5E%7B%5Cfrac%7Bi%7D%7B%5Chbar%7D(p_xx%2Bp_zz)%7D%5Cchi(y)%2C%5C%20%E4%BB%A3%E5%9B%9E%7B%5Crm%20Schrodinger%7D%E6%96%B9%E7%A8%8B%2C$

$%5Cchi%5E%7B%5Cprime%5Cprime%7D(y)%2B%5Cfrac%7B2m%7D%7B%5Chbar%5E2%7D%5B(E%2B%5Cmu%20s_zB-%5Cfrac%7Bp%5E2_z%7D%7B2m%7D)-%5Cfrac%7B1%7D%7B2%7Dm%5Comega_c%5E2(y-y_0)%5E2%5D%5Cchi(y)%3D0%2C%5C%20$

其中 $y_0%3D-%5Cfrac%7Bcp_x%7D%7BeB%7D%2C%5C%20%5Comega_c%3D%5Cfrac%7BeB%7D%7Bmc%7D.$

同一维谐振子的方程类似，解得，

$E_n%3D(n%2B%5Cfrac%7B1%7D%7B2%7D)%5Chbar%5Comega_c%2B%5Cfrac%7Bp_z%5E2%7D%7B2m%7D-%5Cmu%20s_zB%2C$

$%5Cchi_n(y)%3D%5Cfrac%7B1%7D%7B%5Cpi%5E%7B%5Cfrac%7B1%7D%7B4%7D%7Da%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%5Csqrt%7B2%5Enn!%7D%7De%5E%7B-%5Cfrac%7B(y-y_0)%5E2%7D%7B2a%5E2%7D%7DH_n(%5Cfrac%7By-y_0%7D%7Ba%7D)%2C$

其中 $a%3D%5Cfrac%7B%5Chbar%7D%7Bm%5Comega_c%7D%2C%5C%20H_n$ 是Hermite Polynomials. 考虑x方向运动不受限，则 $p_x$ 取连续值. 若x方向由 $L_x$ 所限，由箱归一化则其取值是分立的，

$p_x%3D%5Cfrac%7B2%5Cpi%5Chbar%7D%7BL_x%7Dl%2C%5C%20l%3D0%2C%5Cpm1%2C%5Cpm2%2C%5Ccdots.$ 相对应的平衡位置也是分离的 $%5CDelta%20y_0%3D%5Cfrac%7Bc%7D%7BeB%7D%5Cfrac%7B2%5Cpi%5Chbar%7D%7BL_x%7D.$