What is the Minimum Energy of a One-Dimensional Linear Oscillator?

[Abdul.119]

Homework Statement

The (classical) energy of one-dimensional linear oscillator is

a) show, using the uncertainty relation, that the energy can be written as

b) Show that the minimum energy of the oscillator is

Where

Homework Equations

Δp Δx >= ħ/2
p ≈ ħ/2x

The Attempt at a Solution

I'm really stuck on this problem, I could say that the p_x = ħ^2 / (2x)^2 then substitute in the equation, but still where did the π and the 32 come from? I couldn't find any other relevant equations
Edit: Oh I just realized ħ = h/2π , then p^2 = h^2/(4πx)^2 , substituting in the equation would give the answer.
But now what about b)? I believe to find the minimum energy we set Δx >= ħ/2Δp , but it isn't clear to me how the frequency and angular velocity come in

 

[Orodruin]

Are you familiar with the difference between h and ħ?

 

[Abdul.119]

Are you familiar with the difference between h and ħ?

Oh yes, I just realized that a minute ago and edited my post, but now I'm still having trouble with part b)

 

[Orodruin]

Are you familiar with how to find the minimum of a function?

 

[Abdul.119]

Are you familiar with how to find the minimum of a function?

Hmm I believe I have to take the derivative with respect to Δp, minimum energy dE/dΔp = 0

 

[Orodruin]

Δp is no longer part of your expression for the energy ... You have just expressed it as a function of x only.

 

[Abdul.119]

Then the derivative of p_x = ħ^2 / (2x)^2 ?

 

[Orodruin]

No, it is the energy you wish to minimise.

 

[Abdul.119]

use dE/dx = 0 ?

 

[Orodruin]

use dE/dx = 0 ?

Why don't you try it and see what you get?

 

[Abdul.119]

Why don't you try it and see what you get?

Would get
E = -2h^2/32mx^3π^2 + 2Cx
Doesn't make sense because I still don't see where the omega and f come from

 

[Orodruin]

You care far too much about the omega and f. You are just supposed to find the minimum energy, which is a certain given expression in terms of omega where omega is an expression in other variables which arethe actual variables you have.

 

[Abdul.119]

You care far too much about the omega and f. You are just supposed to find the minimum energy, which is a certain given expression in terms of omega where omega is an expression in other variables which arethe actual variables you have.

So you mean E = -2h^2/32mx^3π^2 + 2Cx can be simplified into

? how?

 

[Orodruin]

No, that is your energy derivative. You need to put it to zero to find the minimum.