- #1

Uniquebum

- 55

- 1

## Homework Statement

Determine the energy uncertainty [itex]\Delta E = \sqrt{<E^2> - <E>^2}[/itex] for a particle described by a wave function

[itex]\Psi (x) = c_1 \psi (x)_1 + c_2 \psi (x)_2[/itex]

where [itex]\psi_1[/itex] and [itex]\psi_2[/itex] are different (orthonormal) energy eigenstates with eigenvalues [itex]E_1[/itex] and [itex]E_2[/itex].

## Homework Equations

I'd presume you need to know

[itex]\hat E = i \hbar \frac{\partial}{\partial t}[/itex]

[itex]\hat E_{kin} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}[/itex]

## The Attempt at a Solution

First, I'm not sure whether i should throw in the kinetic or total energy operator. If i put in the total energy operator, i'll have to derivate the function in respect to time which in this case would result in 0. If i put in the kinetic energy operator it just might work but I'm not sure how i work with those expectation values when they're of the form <E^2> or <E>^2.

Assuming i'd use the total energy operator, should it look like

[itex]<E^2> = \int_{-\infty}^{\infty} \psi^* (x) i^2 \hbar^2 \frac{\partial^2}{\partial t^2} \psi (x)[/itex]

[itex]<E>^2 = \int_{-\infty}^{\infty} \psi^* (x)^2 i^2 \hbar^4 \frac{\partial^2}{\partial t^2} \psi (x)^2[/itex] ?

Any help would be appreciated.