(Solved): Problem 1. Change of basis in second quantization: We can represent quantum operators in different ...
Problem 1. Change of basis in second quantization: We can represent quantum operators in different basis when we use the first quantized expression. We have formulated second quantization during the lecture. Let's consider the second quantized operator for kinetic energy for free particles in a one dimensional box with length
L. That is,
widehat(H)_(kin)=\sum_k ,k^(')a_(k)^(†){k|(widehat(P)^(2))/(2m)|k^(')}a_(k^('))=\sum_k ((?k)^(2))/(2m)widehat(n)_(k)Here, we use
|kto represent the single particle ket for eigen basis of momentum operator.
widehat(P)|kand
{x|k}=(1)/(\sqrt(L))e^(ikx). Suppose we need to change the basis into a basis formed by
|\alpha . Here,
{x|\alpha }=f_(\alpha )(x).,(\alpha _(k)^(†))/(a_(k))are the rising and lowering operator for the single particle mode
k. How to change the basis to express
|kusing the basis
|\alpha in first quantization ex- pression? Express your result using
f_(\alpha )(x)and required operations. (Hint: we want to express our result where
xis considered as a dummy index. We can do that by inserting compete set
{:\sum_(\alpha ) |\alpha }{(\alpha |):}first. Then, insert complete set to express
{\alpha |k}.) How to change the basis of a second quantized Hamiltonian? Find
c_(\alpha )(k)in
a_(k)^(†)=
\sum_(\alpha ) c_(\alpha )(k)a_(\alpha )^(†). (Hint: We can find it by comparing
widehat(H)_(kin)=\sum_(\alpha ) ,\alpha ^(')a_(\alpha )^(†){\alpha |(widehat(P)^(2))/(2m)|\alpha ^(')}a_(\alpha ^('))and the change of basis process in the previous question.) Express the second quantized operator
widehat(H)_(kin)in the real space representation.
