Quantum Mechanics homework question on wavefunctions

[jqmhelios]

Homework Statement

A particle moves in one dimension subject to a potential that is zero in the region
−a ≤ x ≤ a and infinite elsewhere. At a certain time its wavefunction is
ψ = (5a)^(−1/2) cos(πx/2a)+ 2(5a)^(−1/2)sin(πx/a)
.
What are the possible results of the measurement of the energy of this system and
what are their corresponding relative probabilities? What are the possible forms of
the wavefunction immediately after such a measurement? If the energy is immediately
remeasurred, what will now be relative probabilities of the possible outcomes?

Relevant Equations

I assume the solution will involve (H^)f= Ef, the eigenvalue equation for the Hamiltonian operator. Probability should be square of the modulus of the values of an

Absolutely no clue on how to even begin this question due to the exceptionally poor quality of our lectures, who has also flatly refused to give out any solutions, which I could have used to understand what is going on.
I assume the energy has to be obtained by using the eigenfunction equation for a Hamiltonian operator.
The probability should be modulus of the square of a constants, but I do not have the faintest idea on how to obtain that as well.

 

[anuttarasammyak]

You should get energy eigenfunction of this infinite square well first. Then expand the wave function by these eigenfunction bases. Square of absolute value of expansion coefficient gives probability of finding that energy eigenstate in energy observation.
ref. https://en.wikipedia.org/wiki/Particle_in_a_box
In a glance,
The first term : ground state
The second term : first excited state

 

[PeroK]

Homework Statement: A particle moves in one dimension subject to a potential that is zero in the region
−a ≤ x ≤ a and infinite elsewhere. At a certain time its wavefunction is
ψ = (5a)^(−1/2) cos(πx/2a)+ 2(5a)^(−1/2)sin(πx/a)
.
What are the possible results of the measurement of the energy of this system and
what are their corresponding relative probabilities? What are the possible forms of
the wavefunction immediately after such a measurement? If the energy is immediately
remeasurred, what will now be relative probabilities of the possible outcomes?
Relevant Equations: I assume the solution will involve (H^)f= Ef, the eigenvalue equation for the Hamiltonian operator. Probability should be square of the modulus of the values of an

Absolutely no clue on how to even begin this question due to the exceptionally poor quality of our lectures, who has also flatly refused to give out any solutions, which I could have used to understand what is going on.
I assume the energy has to be obtained by using the eigenfunction equation for a Hamiltonian operator.
The probability should be modulus of the square of a constants, but I do not have the faintest idea on how to obtain that as well.

You should get a textbook, such as Griffiths, which is good for wave mechanics.

Note that the guidelines on getting homework help preclude you simply saying you haven't a clue. See the Help pages.

 

[PeroK]

Alternatively, there are comprehensive course notes for Infinite Square Well online. E.g. the MIT OpenCourseWare.

 

[Ishika_96_sparkles]

Statement: Particle in a box problem

Here is the usual outline to proceed towards the solution.

Step 1: Measurement of the energy eigenvalues of this system
The potential ##V(x)## is 0 between ##x=-a## and ##x=a##. Elsewhere it is infinite. See the picture above.
Write the time-independent Schroedinger's equation for this case, within the given limits ##-a \leq x \leq a##
$$\frac{\partial^2\psi}{\partial x^2} + \big( \frac{2m}{\hbar}\big) E \psi =0$$
rewrite this equation using ##\big( \frac{2mE}{\hbar}\big)=k^2## such that we get the familiar linear second order homogeneous differential equation
$$\frac{d^2\psi}{d x^2} + k^2 \psi =0$$
Step 2: Write the general solution for this equation i.e., ##\psi= A \times \sin( \cdots ) \pm \cos( \cdots ) ## and use the boundary conditions ##-a \leq x \leq a## where the wavefunction vanishes to calculate the value of ##k## plus some phase ##\phi## within the bracket of the sine term i.e., ##\sin( \cdots )##.

Step 3: use ##\big( \frac{2mE}{\hbar}\big)=k^2## to calculate E by putting the value of ##k## on the R.H.S. You have now the energy eigenvalue spectrum.

Step 4: Now construct the eigenfuntions and solve for the unknown constant A by using the normalization condition ##\int^a_{-a} \psi^*\psi dx=1##.

Step 5: Post your work here as a pic of the worksheet. Then ask for further guidance.