# Demostration of the Uncertainty Principle from a given ket

1. Feb 15, 2016

1. The problem statement, all variables and given/known data

I have to demonstrate the Uncertainty Principle

Starting from the expression of the following ket:

|Ψa>=(ax^+ibp^)|Ψ>

where a and b are complex numbers and the ^ denotes that x and p are unitary vectors.

2.Relevant equations

I must use the bra-ket notation, but I don't really have guidelines about the procedure to follow or the equations to use.

3. The attempt at a solution

I thought about solving it trying to demostrate that <Ψa|Ψa> must be equal or greater to zero, but this implies integrate a expresion in two dimenssions of the phase space that I don't know how to solve. I though about using a projection operator, but all my tries went wrong, so I don't know if this is the right procedure.

2. Feb 15, 2016

### BvU

This is quite an involved exercise. You want to sort out what you have available to deal with it, and collect the relevant equations. A bit more than just "bra-ke notation". I'll give you a few to consider:
• The expectation value for any observable $A$
• The expression for $\sigma^2$ (being such an observable)
• The Schwarz inequality
• How your $\Psi_a$ fits in all of this
And then you want to embark on the solution phase; please show your steps. Just thinking and saying "can't do" does not count in PF . Getting stuck is no problem: that's what PF is for. But you can't get stuck if you don't take some steps first.

3. Feb 16, 2016

Thank you BvU. Here is some information about the procedure I followed.

I used the condition for <Ψa|Ψa> having to be equal or greater than zero, and I solved the integral. I did the substitution of the operators x and p, and I have the following expression:

<Ψa|Ψa>= a^2<x^2>+b^2<p^2>+∫Ψ*[x,p]ΨdV

Where the terms between brackets <> define expected values and the term in [] is a commutator.

Someone knows if I can obtain the expression of the uncertainty principle with this, demonstrating that it has to be greater than zero?

Last edited: Feb 16, 2016
4. Feb 16, 2016

### BvU

How can you manipulate things so you end up with a $\sigma_x$ and a $\sigma_p$ (c.q. their product, or their product squared..) ?
All I see now is an $<x^2>$ and a $<p^2>$ (and an integral where a and b have mysteriously disappeared ).

5. Feb 16, 2016

My fault, the a and b are there, I forgot to include them.

I'll put here the guideline I followed (I'm not sure it's right):

I began with this (where x and p are the operators position and momentum):

<Ψa|Ψa>= ∫Ψ*(ax-ibp)·Ψ(ax+ibp)dV = ∫Ψ*($a^2$·$x^2$+$b^2$·$p^2$+iab(xp-px))Ψ adV = ∫Ψ*($a^2$·$x^2$)ΨdV+
∫Ψ*($b^2$·$p^2$)ΨdV+iab∫Ψ*([x,p])ΨdV

And, the first two ones are the expressions of the expected values of the squared operators, so:

<Ψa|Ψa>= $a^2$·$<x^2>$+$b^2$·$<p^2>$+iab∫Ψ*([x,p])ΨdV

And that's the equation I must equal to zero to obtain de Uncertainty Principle. It's correct? In that case, some ideas of how can I follow, or any guidelines to do it another way if this is wrong?

Thank you for your time and the patience with a novice.

6. Feb 16, 2016

### BvU

I'm not much better at the Heisenberg stuff than any novice (brought up with the Schroedinger picture) - which is why I'm following this with great interest (I'm learning too !).

I get the idea you forget $a$ and $b$ are complex numbers ?
Anyway, I see the $\hbar$ appearing from $[x,p] = i\hbar$, but what about the various $\sigma$ ?

Don't think so: what does that have to do with $\sigma_x \sigma_p$ ?

7. Feb 16, 2016

Yes, a and b are complex, but I think that doesn't interferes with the calculus I made, no?

I can't see neither the correlation between σxσp and my result, but the tip they give to us is that it can be made with the condition <Ψa|Ψa>=>0. and that's the only expression I could reach for that. Maybe there's another way to get another result, but I didn't find it yet.

8. Feb 16, 2016

### BvU

Think of an expression for $\sigma_x^2$. Idem $\sigma_p^2$. And their product (or the square root thereof).

9. Feb 16, 2016