Recent content by MFAHH

Perfect! Many thanks for your time and help. I think I've got this down now :)

Final thing, for ##⟨ψ|X^2|ψ⟩## I got: ##⟨ψ|X^2|ψ⟩= \frac{h}{mw} ##, is that correct?

Aha! ##⟨ψ|X|ψ⟩= \sqrt{\frac{h}{2mw}} cos(wt) ## And so: ##⟨ψ|X|ψ⟩^2= \frac{h}{2mw} cos^2(wt) ## Now it looks splendid.

Yes, I should've clicked on. It would simplify to: ##⟨ψ|X|ψ⟩= \sqrt{\frac{h}{2mw}} cos(\frac{t}{h} (E_0 - E_1)) ## And so: ##⟨ψ|X|ψ⟩^2= \frac{h}{2mw} cos^2(\frac{t}{h} (E_0 - E_1)) ##

Great! So I've calculated ## ⟨ψ|X|ψ⟩ ## and obtained: ##⟨ψ|X|ψ⟩= \sqrt{\frac{h}{8mw}} (e^{\frac{it(E_0 - E_1)}{h}} + e^{\frac{it(E_1 - E_0)}{h}}) ## As for part c, for ## ⟨ψ|X|ψ⟩^2 ## I square the expression above to get: ##⟨ψ|X|ψ⟩^2 = \frac{h}{8mw} (e^{\frac{2it(E_0 - E_1)}{h}} +...

Good point :). For the first part of b) I've got the following, just want to check it's right. ##|ψ⟩ = e^{\frac{-iE_0t}{h}}\frac{1}{\sqrt{2}}|0⟩ + e^{\frac{-iE_1t}{h}}\frac{1}{\sqrt{2}}|1⟩ ## where ## E_n ## is the nth energy eigenvalue given by ##E_n = (n+\frac{1}{2})hw##

Is it worth putting ##\theta## in or considering the simplest case for the following parts do you think?

Ah I see now, so it's the case that: ##|ψ⟩ = \frac{1}{\sqrt{2}}|0⟩ + \frac{1}{\sqrt{2}}|1⟩ ##

Ah oops, is it: <X> = (h/2mw)1/2 [A*B + B*A] I ended up with α = π/4. And so |ψ⟩ = A|0⟩ + B|1⟩ = cos(π/4) |0⟩ + sin(π/4) |1⟩ Not sure about that part as we've solved for the real parts of A and B, but not A and B themselves.

I've ended up with: <X> = (h/2mw)2 [A*B + B*A] Now to obtain |ψ⟩ which maximizes <X>, we set <X> = 0 and solve for A and B. This gives: A*B = -B*A. How would one solve for the two unknowns then?

Thanks for the reply. Ok so I substituted |ψ⟩ = A|0⟩ + B|1⟩ into ⟨ψ|X|ψ⟩, and from simple manipulation I have: <X> = A*A⟨0|X|0⟩ + A*B⟨0|X|1⟩ + B*A ⟨1|X|0⟩ + B*B ⟨1|X|1⟩ Is this correct? I wonder what can be done to simplify it, should I substitute X in terms of a and a†?

Homework Statement Consider a linear harmonic oscillator with the solution defined by the ladder operators a and a†. Use the number basis |n⟩ to do the following. a) Construct a linear combination of |0⟩ and |1⟩ to form a state |ψ⟩ such that ⟨ψ|X|ψ⟩ is as large as possible. b) Suppose that...

Anybody? :)

:oldconfused:

Awesome, so from what I read there I'm more or less on the right track. As for the next part of the question, how is it that I am meant to proceed? Is it just that I rearrange the young Laplace equation for the pressures above and below? But then that answer won't be in terms of what I know from...