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(Solved): The quantum-mechanical harmonic oscillator Hamiltonian is given by H=(1)/(2m)p^(2)+(1)/(2)m\omega ^( ...
The quantum-mechanical harmonic oscillator Hamiltonian is given by H=(1)/(2m)p^(2)+(1)/(2)m\omega ^(2)x^(2).
(a) The ladder operators a,a^(†) are given by the expressions
a=\sqrt((m\omega )/(2ℏ))x+(i)/(\sqrt(2ℏm\omega ))p
a^(†)=\sqrt((m\omega )/(2ℏ))x-(i)/(\sqrt(2ℏm\omega ))p
(a) First show that [a,a^(†)]=1. Then derive the expression for the Hamiltonian in terms of the ladder
operators, H=ℏ\omega (a^(†)a+(1)/(2)).
(b) The states |n: form an orthonormal set, and have the properties
a|n:
Find the harmonic oscillator spectrum E_(n).
(c) Calculate the following expressions:
(:n|x|n:),(:n|p|n:),(:n|x^(2)|n:),(:n|p^(2)|n:),(:n|x^(101)|n:).
\int_(-\infty )^(\infty ) dxe^(ikx)=2\pi \delta (k),[x,p]=iℏ,\int_0^(\infty ) d\times ^(2)e^(-ax)=(2)/(a^(3)),a>0,\int_0^(\infty ) d\times ^(4)e^(-ax)=(24)/(a^(5)),a>0