- #1
LizardWizard
- 18
- 0
For a 1D QHO we are given have function for ##t=0## and we are asked for expectation and variance of P at some time t.
##|\psi>=(1/\sqrt 2)(|n>+|n+1>)## Where n is an integer
So my idea was to use Dirac operators ##\hat a## and ##\hat a^\dagger## and so I get the following solution
##<\hat P>=<\psi|\hat P|\psi>=i \sqrt {m\omega\hbar/2} <\psi|(\hat a^\dagger - \hat a^)|\psi>=##
##=i \sqrt {m\omega\hbar/2}(1/\sqrt 2) (1/\sqrt 2)(<n|+<n+1|(\hat a^\dagger-\hat a)|n>+|n+1>)##
Knowing that
##\hat a^\dagger |\psi_n>= \sqrt {n+1} |\psi_{n+1}>##
##\hat a |\psi_n>= \sqrt {n} |\psi_{n-1}>##
I eventually arrived at
##<\hat P>=i\sqrt{m\omega\hbar/8}(-\sqrt n <n|n> + \sqrt{n+1} <n+1|n+1>)##
However the answer is supposed to be 0, did I do something wrong or could I perhaps make use of the fact that ##\sqrt{n+1}-\sqrt n ## for ##n\to\infty## equal to 0?
##|\psi>=(1/\sqrt 2)(|n>+|n+1>)## Where n is an integer
So my idea was to use Dirac operators ##\hat a## and ##\hat a^\dagger## and so I get the following solution
##<\hat P>=<\psi|\hat P|\psi>=i \sqrt {m\omega\hbar/2} <\psi|(\hat a^\dagger - \hat a^)|\psi>=##
##=i \sqrt {m\omega\hbar/2}(1/\sqrt 2) (1/\sqrt 2)(<n|+<n+1|(\hat a^\dagger-\hat a)|n>+|n+1>)##
Knowing that
##\hat a^\dagger |\psi_n>= \sqrt {n+1} |\psi_{n+1}>##
##\hat a |\psi_n>= \sqrt {n} |\psi_{n-1}>##
I eventually arrived at
##<\hat P>=i\sqrt{m\omega\hbar/8}(-\sqrt n <n|n> + \sqrt{n+1} <n+1|n+1>)##
However the answer is supposed to be 0, did I do something wrong or could I perhaps make use of the fact that ##\sqrt{n+1}-\sqrt n ## for ##n\to\infty## equal to 0?