1- calculate the matrix (m|p²|n > and (m|x²|n > then find the matrix of kinetic and potential energy of SHM

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in quantum mechanics 

 

1- calculate the matrix (m|p2|n > and (m|x2|n > then find the matrix of kinetic and potential
energy of SHM
Transcribed Image Text:1- calculate the matrix (m|p2|n > and (m|x2|n > then find the matrix of kinetic and potential energy of SHM
Expert Solution
Step 1

The position and momentum operator in terms of ladder operator are defined as : 

xx^=ħ2mωa+app^=iħmω2aax2=ħ2mωa+aa+a=ħ2mωaa+aa+aa+aap2=-ħmω2a-aa-a=-ħmω2aa-aa-aa+aa

The definitions of ladder operators are : 

a|n>=n+1|(n+1)>a|n>=n|(n-1)>

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