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# # problems in Quantum Mechanics

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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

https://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.

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