qit.ho.position

qit.ho.position(n=30)

Position operator.

Parameters

n (int) – truncation dimension

Returns

position operator

Return type

array[complex]

Returns the n-dimensional truncation of the dimensionless position operator Q in the number basis.

$$\begin{array}{rl}Q& =\sqrt{\frac{m\omega }{\hslash }}q=(a+a^{†})/\sqrt{2},\\ P& =\sqrt{\frac{1}{m\hslash \omega }}p=-i(a-a^{†})/\sqrt{2}.\end{array}$$

(Equivalently, $a=(Q+iP)/\sqrt{2}$). These operators fulfill $[q,p]=i\hslash ,[Q,P]=i$. The Hamiltonian of the harmonic oscillator is

$$H=\frac{p^2}{2m}+\frac{1}{2}m{\omega }^2{q}^2=\frac{1}{2}\hslash \omega (P^2+Q^2)=\hslash \omega (a^{†}a+\frac{1}{2}).$$