Calculating Probability of Particle in First Excited State

[oddiseas]

Homework Statement

A particle is in a quantum state defined by:

\Phi(x)=0.917\Psi_1+0.316\Psi_2+0.224\Psi_3+a\Psi_4

where \Psi are the eigenfunctions for a particle in a box given by \Psi_n=\sqrt{2/L}sin(npix/L).

The corresponding eigenenergies are E_n=1.5n^2eV

What is the probability that an energy measurement will find the particle in its first excited state?

Homework Equations

i was thinking to use the integral of the initial state, multiplied by the eigenstate with the energy corresponding to the first excited state, but i am not really sure, it is more of a guess, so if someone could explain the logic to me it would ve appreciated. i always get stuck on the probability questions when they refer to the probability of specific energy states or momentum states> so i would like to understand this concept instead of just copying a method.

The Attempt at a Solution

 

[dextercioby]

There's an axiom (some people call it the <Born rule>) telling you exactly what you need to find out: the probability of getting E_1 when measuring the energy. Search for it in your lecture notes.