Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if P^and Q^have a complete set of common eigenfunctions, then [P^.Q^]f=0for any function in Hilbert space.

Short Answer

Expert verified

A whole set of common eigenfunctions cannot be shared by two noncommuting operators.

Step by step solution

01

Concept used

The same complete set of common eigenfunctions:

P^fn=λnfnandQ^fn=μnfn

02

Calculation

Assuming the operators P^ and Q^have the same complete set of common eigenfunctions, that is:

P^fn=λnfnandQ^fn=μnfn

And suppose the set fnis complete, so that any function in Hilbert space f{x)can be expressed as a linear combination, that is:

f=cnfn

Solving for the above function,

P^,Q^f=P^Q^-Q^P^cnfn=P^cnμnfn-Q^cnλnfn=cnμnλnfn-cnλnμnfn=0

If two operators have the same set of eigenfunctions, the commutator will be zero.

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