Superposition of Eigenfunctions: Probability and Expectation Values

[sarabellum02]

I know this question isn't supposed to be hard but I can't figure it out for the life of me.

If a certain wavefunction is made by superposition of three eigenfunctions of the momentum operator (F1, F2, and F3): wavefunction=0.465F1+0.357F2+0.810F3. The eigenvalues of those eigenfunctions are f1=+0.10, f2=-0.47, and f3=+0.35. What is the probability of a single measurement giving a momentum of +0.10? What is the probability of a single measurement giving a momentum of -0.20? and What is the expectation value of the momentum of the particle?

 

[dextercioby]

Apply the 3-rd principle and the definition of expectation value.

Daniel.

 

[Doc Al]

What is the probability of a single measurement giving a momentum of +0.10?

The probability of a getting a particular eigenvalue when making a measurement is proportional to the (complex) square of the coefficient for that eigenfunction in the wavefunction:
$$\Psi = C_1 F_1 + C_2 F_2 + C_3 F_3$$
Assuming the wavefunction is normalized (as is the one in this example), then the probability of obtaining a value of f1 is ${C_1}^*C_1$.

What is the probability of a single measurement giving a momentum of -0.20?

The only possible values for a measurement are the eigenvalues associated with eigenfunctions that appear in the wavefunction (with non-zero coefficients).

and What is the expectation value of the momentum of the particle?

The expectation value is the weighted average of all possible measurements:
$$<p> = {C_1}^*C_1 f_1 + {C_2}^*C_2 f_2 + {C_3}^*C_3 f_3$$