Uncertainty Principle and Operator Algebra

[noblegas]

Homework Statement

The Heisenberg uncertainty principle can be derived by operator algebra , as follows. Consider a one-dimensional system, with position and momentum observables x and p. The goal is to find the minimum possible uncertainties in the predicted values of the position and momentum in any state |\varphi> of the system. We need the following preliminaries.

Homework Equations

The Attempt at a Solution

a) Suppose the self-adjoint observables q and r satisfy the commutation relation

[r,q]=iq, where c is a constant(not an operator). Show c is real.

should I take the self-adjoint of [r,q], i.e.[r,q]^{\dagger} ?

b) Let the system have the normalized state vector |\varphi> and define the ket vector

|\phi>=(\alpha*r+iq)|\varphi> where \alpha is a real constant(again, a number , not an operator). used equations <\phi|\phi> >=0 and [r,q]=ic to show that

\alpha^2<r^2>-\alpha*c+<q^2>>=0, where <r^2>=<\varphi|r^2|\varphi> and q^2

Should I begin by finding <\phi|\phi>?

Since, |\phi>=(\alpha*r+iq)|\varphi> would that mean <\phi|=<|\varphi(\alpha*r-iq)

c) By seeking the value of \alpha that minimizes the left side of the equation \alpha^2<r^2>-\alpha*c+<q^2>>=0, show

<r^2><q^2>=c^2/4

should I multiply the expectation value <r^2> to the equation \alpha^2<r^2> -\alpha*c+<q^2>>=0

 

[noblegas]

No one understood my question and/or solution?