Quantum Optics II: QM Intro & the QHO

The first thing you should be familiar with is Schrodinger’s equation:



Here i is the sqrt(-1), ħ is Planck’s constant divided by 2π, ∂/∂t is the partial time derivative, and H is called the Hamiltonian. The Hamiltonian is an “operator”. It operates on whatever is to the right of it, which is our wave function φ . The wave function contains everything we need to know about the system.

As in classical mechanics, the Hamiltonian can usually be written as the sum of the kinetic and potential energies. When written for a particular system, H will contain spatial derivatives and variables. Sometimes it’ll contain time variables too, but we treat the time-dependent Hamiltonian differently than what I’m going to present here.

There are lots of different operators in QM and they have special and convenient properties. We won’t concern ourselves much with all of these subtle points and definitions though- grab an intro to quantum book if you’re interested.


Equation 1 is a partial differential equation. It alone is not enough to solve the problem, you also need boundary conditions that tell you the value of φ at certain times or locations.
I’ll also mention that if we integrate φ* φ (where the * denotes the complex conjugate) and integrate over all space, we get 1. That’s because φ* φ is a probability density function. If we want to know the probability the particle is between x1 and x2 (in a one dimensional system), we would just calculate:



And if x1 and x2 are ±∞, the probability is unity. The particle has to be somewhere.

Sometimes we write in terms of things that make physical, intuitive sense, like position coordinates (x,y,z) and time (t). This is often mathematically cumbersome. Schrodinger’s approach to QM is nice because it’s easy to think in terms of location and time. However, partial differential equations are difficult to solve analytically except for a few special cases. So Heisenberg developed a matrix approach to quantum mechanics that uses linear algebra. This is better, but writing down huge matrices and vectors can be cumbersome (especially on this blog… my god I couldn't even format equations). So Dirac invented his own notation that is just the best thing ever. I’ll be using it a lot.

In short, Dirac notation is just a clean way of writing down matrix equations which are an alternative to differential equations. They all say the same thing at the end of the day.

Using this alternative notation, Schrodinger’s equation becomes:



It’s identical except that we put these weird brackets around the wave function. Don’t be fooled, the devil is always in the details. The wave function is now called a state vector and the Hamiltonian is a matrix of dimensions that correspond to the dimensions of the state vector. In reality this equation is still filled with derivatives and position and sometimes time variables, but we rarely think of it in those terms. Usually H just has some simple effect on the state vector that allows us to write down the solution to the Schrodinger equation without thinking much.
This might sound confusing now, but it’ll be clearer when we do an example.


As I said earlier, in the problems we’ll encounter H rarely contains time variables or derivatives, we call these “stationary” problems. We don’t care about time as much as we care about space in these systems. If H does not depend on time, we can write the Schrodinger equation as:



Now, Eq. (3) is an “eigenvalue” problem. The solution is a linear combination of many “eigenfunctions” or “eigenvectors” (if we’re talking about the Schrodinger or Heisenberg formalism, respectively) that correspond to an eigenvalue. There will be a lot of similar solutions that satisfy Eq. (3) and its boundary conditions, and each of these individual solutions is identifiable by its eigenvalue. In Eq. (3) we called the eigenvalue E, because E happens to be the energy in the case of a time independent Hamiltonian (since H=kinetic+potential energies).
If we wanted to talk about a single component of φ, we could denote it φ_n or simply |n>, where n denotes the nth eigenvalue, E_n. Some systems have lots of different eigenvalues, some just have one.


Continuing on… for a time independent Hamiltonian, the general solution of the Schrodinger equation is a sum of all these possible eigenvectors with some coefficients:



C_n is just a constant and the exponential comes from the time derivative in Eq (1). If we don’t care about time, this is just another constant number (just say t=0 and its 1). This looks like a simple answer, but finding the expressions for |n> in terms of physically useful variables, like position, is nontrivial. That’s why we’re just using Dirac notation so we can just call it |n> .

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Let’s go ahead and do a simple example that illustrates the power of operators and Dirac notation.

The Quantum Harmonic Oscillator
+ Show Spoiler +
This is a very beautiful problem, and you encounter it several times in QM. I hardly did it justice here, so if you want to look at the problem in all of its glory, (like how to solve the differential equation and cool properties of ladder operators) I recommend you visit google or a textbook if you have one around.

Hopefully that gives you a feeling for what quantum looks like. Everything comes from complicated differential equations, but some very smart people have found clever ways to translate them into matrix equations which are easier to solve. They also did us the favor of defining a few cool operators that magically give us really simple answers when they act on the eigenfunctions of the system, like using a† and a to move our system up and down in energy levels.


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I get the feeling that this was maybe a boring read, and if so I apologize for that. We just needed to establish some sort of foundation so when I start using math to describe interesting things you guys are familiar with my notation.
Is this level of math and explanation okay? Keep in mind many optics lessons will be less equation heavy and more concept-focused. The idea of this blog post was just to get a feel for what level math I can use here and introduce some basic ideas that are necessary to speak quantum.