Given the uncertainty in momentum, calculate the uncertainty in Energy

[datran]

Homework Statement

I am given that E = (hbar)^2 k^2 / 2m. I am suppose to calculate the uncertainty in energy(ΔE).

I am also given the wave equation.

I am given Δp (uncertainty in p), Ko (wave number). and that Δp << hbar Ko.

Homework Equations

I believe the equation is ΔE = sqrt(<E^2> - <E>^2).

The Attempt at a Solution

So I found <E> = <p^2> / 2m. How am I suppose to find <E^2>? I tried to just square the given E formula and then find <E^2> that way. However, I get <hbar^4 k^4 > / 4m^2. I do not know what to do from there.

Any help would be great. Thanks!

 

[mark726]

Hello,

To calculate <E^2>, you can use the wave equation and the given formula for E. First, plug in the given E formula into the wave equation to obtain:

Ψ = A * exp(i(kx - ωt))

Where A is the amplitude, k is the wave number, and ω is the angular frequency. Now, to find <E^2>, we need to take the expectation value of E^2, which is:

<E^2> = ∫ Ψ*E^2 Ψ dx

Using the given formula for E, we can simplify this to:

<E^2> = (∫ Ψ* (hbar)^2 k^2 / 2m Ψ dx)^2

= (hbar)^4 k^4 / 4m^2 * (∫ Ψ* Ψ dx)^2

= (hbar)^4 k^4 / 4m^2 * (Ψ* Ψ)^2

= (hbar)^4 k^4 / 4m^2 * A^4

= (hbar)^4 k^4 / 4m^2 * |A|^4

= (hbar)^4 k^4 / 4m^2 * (|A|^2)^2

= (hbar)^4 k^4 / 4m^2 * 1

= (hbar)^4 k^4 / 4m^2

Therefore, <E^2> = (hbar)^4 k^4 / 4m^2. Now, using this and the given formula for <E>, you can plug these values into the formula for ΔE and solve for the uncertainty in energy. I hope this helps! Let me know if you have any further questions.