Have you ever wondered what it would be like to study quantum physics at a prestigious university ?
In a few seconds, I will unbox a quantum physics exam from MIT.
But first I wanted to show you that I've got a new kitten. His name is ketchup.
The exam I'll show is publicly available on the MIT Open course website.
The link will be down in the description and it's from the course quantum physics 1 which is an undergraduate course.
The website also has some lecture notes and even an online version of the course
which you can take which would help you learn the material that's involved here.
But for now,
Let's just have a look at what the exam involves.
It will seem difficult for anyone who hasn't taken this exact course.
So don't let it discourage you and
Take it with a bit of a grain of salt if you're just looking at this from the outside.
Here's the exam here there are six questions and we've got three hours to do it.
First, Let's have a little look at the formula sheet which comes with the exam.
Have a bit of everything on here
Fourier transforms, Delta functions, things about the Schrodinger equation
and operators as well as some integrals.
A solution of the Schrodinger equation for an infinite square well.
And over here, maybe some things for more specific questions.
We have the raising and lowering operators.
We have things to do with normalization and angular momentum.
We won't use all of these things during the exam.
This is kind of just the set of all possible things which we might need to remember here.
Opening up the exam,
Our first question has to do with
Eigenvalues and Eigenfunctions for this strange-looking potential that we have here.
So as a bit of context in quantum physics, you have something called a wave function which
contains all the measurable information about a system.
If you operate something called the Hamiltonian which has to do with total energy on the wave function,
we get the Schrodinger equation
and solutions for the time independent
Schrodinger equation only exists for certain values of energy and these
energy values are called Eigenvalues.
These Eigenvalues have associated Eigenfunctions and this question is asking us various properties about these
when we have this particular like potential energy.
Going on,
We have something about the double slit experiment where we get this pattern on the wall.
We're asked to work out something about the set up.
Another question here about wave functions and potentials.
So, it's saying if we have this strange-looking potential shape here,
What is our wave function going to look like?
And they've drawn this picture but there's something wrong with this picture.
And the question is asking us to work out
Why this fails to be a correct representation?
This page here is similar again.
It's about boundary conditions this time and still about Eigenstates.
Notice that we're still on question one
which is upto G and H right here.
Another similar one here
except this time we're being asked to draw the probability density
or essentially the wave function squared.
And then a question asking us how solutions would be different for
bosons, fermions or distinguishable particles.
That's finally the end of question 1
and question 2 here starts off with actually a bit of condensed matter physics.
So we're asked about the link between the letter spacing and diamond and the band gap.
as the letter spacing is decreasing
the atoms are closer together and the binding force between the valence electrons and the parent atoms will increase.
For this we would expect the band gap to go up because the band gap is
representing the amount of energy that needs to be given to these valence electrons to make them become conduction electrons.
Another question here about diamond
and it's asking what is the minimum wavelength at which a diamond in a jewelry store is opaque.
On to question 3 and we're being asked to consider a mass m
and 2d confined to a square box by an infinite potential.
And then there is no potential inside the box and we asked several questions about it.
This is probably somewhat standard for questions about quantum physics.
Things like being asked about the degeneracy of states
and quantum physics,
States are degenerate if they give the same value of energy upon measurement.
Question four is all about the hydrogen atom.
We have some questions here about
normalization of the wavefunction,
How the wavefunction will evolve with time,
And then a few problems about expectation value.
The expectation value is the expected value of the result of a measurement of an experiment.
It's an average of all the possible outcomes of the experiment weighted by their likelihood.
So it's a bit of statistics
and there's a few questions here about it.
Expectation value is denoted by these brackets here.
Some more questions about making a measurement in terms of wave functions and then question five.
We're given a potential here.
We're told, "Suppose the potential has precisely two bound states. Sketch the corresponding wave functions.
How does the higher energy bound state look when its energy is close to zero? Do another sketch of that. "
So a bit of drawing.
And then answering more questions about that little system.
Here we have a wave that is being reflected.
So here we're asked to calculate the reflection coefficient
which will involve finding out C and D from this equation here.
This is a general solution to the Schrodinger when we solve it using
various boundary conditions that we know of.
On to our final question number six
and it's about the harmonic oscillator
Here, we can see that we're using the letter operators.
These operators can raise or lower the
eigenvalue of another operator.
So in terms of energy
They could go from one energy level to the next
and I think they will be a useful tool in answering some of these questions here.
We're being asked to find the allowed energies for a particle in a particular harmonic oscillator potential.
We're using our ladder operators here
and some relations between them to show a couple of statements
And a few more over here.
And then we're on to our last question.
Verify that these here are eigenfunctions of this and find their eigen values.
So it looks like to do well in this exam,
you need to know your Schrodinger equation and solving it very well
as well as dealing with eigenvalues and eigenfunctions.
These are kind of abstract terms, but they will be
introduced to you in any first course on quantum mechanics.
I would say this exam was quite tricky.
I don't think I would pass just from my memory of what I did in my own quantum mechanics courses
You really need to be back in the practice of solving problems like this and
remembering how all the details work together.
A lot of people have sort of a hobby interest in quantum mechanics
but this would be quite a hard exam if you haven't done some more of the rigorous
proofs or examples that you would find in a textbook.
So I would say that something like
the Griffiths quantum mechanics textbook would be good level for some of the stuff
and also just checking out, you know, the lecture notes and examples on the MIT course itself.
We saw a lot of working with potentials here.
And just you know being able to adapt our knowledge of potentials to work with some,
you know more unusual shapes.
I hope that was a little bit of insight for you into the world of what it's like to study this somewhere like MIT.
Thanks for watching and I'll see you next time!
