# # problems in Quantum Mechanics

1. show that the commutator obeys:

[A,B] = -[B,A]

[A,B+C]=[A,B] + [A,C]

[A,BC]=[A,B]C+B[A,C]

[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0

Given the fundamental commutator relation between momentum and position [x,p] = ih

show that:

a. [x^n,p] = ihn*x^(n-1)

b. [x,p^n] = ihn*p^(n-1)

c. show that if f(x) can be expanded in polynomial in x and g(p) can be expanded in a polynomial in p, then [f(x),p] = ih*df/dx

and [x,g(p)] = ih * dg/dp

for problem 3 that deals with Hamiltonian, observables, diagonalization and energy values see attached file

© BrainMass Inc. brainmass.com October 9, 2019, 9:58 pm ad1c9bdddfhttps://brainmass.com/physics/energy/problems-quantum-mechanics-203703

#### Solution Preview

See attached files

b.

c.

d.

Before we begin, some more commutation identities:

1.

2.

3.

4.

i.)

We shall use induction to prove that .

We show that for n=1,2,3 this statement is true:

So now we assume it is true for n, and we have to show that if so, it must be true for n+1 as well:

And this concludes the proof (we have shown that it is true for n=3, and therefore must be true for n=4. But then, if it is true for n=4 then it is true for n=5 and so forth and so on)

ii)

The same procedure is used to show that :

For n=1,2,3

And if true for n, then it is true for n+1:

QED

iii)

If the function is a polynomial expansion in ...

#### Solution Summary

The 13 pages file describes in detail how to solve the three problems. The third problem deals with the Hamiltonian operatora nd expectation values of severeal operators.