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?