# Particle in a Box: Energy Measurements and Probabilities Explained

• phrygian

## Homework Statement

A particle is in an infinite square well extending from x = 0 to x = a. Its state is a
linear combination of the two lowest energy states.
Psi(x, t) = A(2 Psi1(x, t) + Psi2(x, t))
a) If a measurement of energy is made what are the possible results of the measurement?
What is the probability associated with each? What is the average value of energy?

## The Attempt at a Solution

I know that when the Hamiltonian operates on Psi_n(x,t) it give E_n Psi_n(x,t), right? So are the possible results just Psi1(x,t) and Psi2(x,t) or would they be 2A E1 Psi1(x,t) and A E2 Psi2(x,t)? I am confused because I know quantum state vectors are supposed to be normalized but I am confused how that translates to wave functions?

And as for the probability, I really am not sure where to start can someone point me in the right direction?

Thanks for the help

The possible outcomes of a measurement are the eigenvalues of the corresponding observable. The coefficients of the wavefunction, written as a superposition of eigenstates, will tell you what those respective probabilities are.

This question is really just a straightforward application of the quantum mechanical rules (postulates) so it's hard to help without giving the answer.

phrygian said:

## Homework Statement

A particle is in an infinite square well extending from x = 0 to x = a. Its state is a
linear combination of the two lowest energy states.
Psi(x, t) = A(2 Psi1(x, t) + Psi2(x, t))
a) If a measurement of energy is made what are the possible results of the measurement?
What is the probability associated with each? What is the average value of energy?

## The Attempt at a Solution

I know that when the Hamiltonian operates on Psi_n(x,t) it give E_n Psi_n(x,t), right? So are the possible results just Psi1(x,t) and Psi2(x,t) or would they be 2A E1 Psi1(x,t) and A E2 Psi2(x,t)? I am confused because I know quantum state vectors are supposed to be normalized but I am confused how that translates to wave functions?

And as for the probability, I really am not sure where to start can someone point me in the right direction?

Thanks for the help

First of all, all this equation states: ''Psi(x, t) = A(2 Psi1(x, t) + Psi2(x, t))'' is two particles oscillating through a specific position and a time. You must state they are in a superpositioning, which is actually a type of interference. The total energy of one of these particles is:

$E=\hbar \omega$. It also has a respective wave number $k$: $\frac{\hbar^{2}k^{2}}{2m}$. Now work it out for two particle systems.

ManyNames said:
First of all, all this equation states: ''Psi(x, t) = A(2 Psi1(x, t) + Psi2(x, t))'' is two particles oscillating through a specific position and a time. You must state they are in a superpositioning, which is actually a type of interference. The total energy of one of these particles is:

$E=\hbar \omega$. It also has a respective wave number $k$: $\frac{\hbar^{2}k^{2}}{2m}$. Now work it out for two particle systems.
The problem statement clearly says the state describes one particle, not two.

vela said:
The problem statement clearly says the state describes one particle, not two.

Then the equation required, the one given is not the ordinary standard equation i have ever seen. This is the one which will calculate the question at hand.

$\psi(x,t) = [A sin(k,t)+B cos(k,t)]e^{-i \omega t}$

ManyNames said:
Then the equation [...] given is not the ordinary standard equation i have ever seen.
Really? This is a standard question in introductory quantum mechanics.

vela said:
Really? This is a standard question in introductory quantum mechanics.

Then why is there a factor of 2 for both psi identities which appear inside the brackets?

What's a psi identity?

$\Psi(x,t)$ isn't an energy eigenstate. It's a superposition of the energy eigenstates $\psi_1(x,t)$ and $\psi_2(x,t)$.