The Creation and Annihilation Operators (collectively known as the Ladder Operators) are a very useful tool in quantum mechanics. We'll be taking a look at what they represent and how we can use them.
Before delving into the world of quantum mechanics, we'll first be looking at an important classical system - a classical harmonic oscillator. This is any system that undergoes simple harmonic motion, where the acceleration of our system is proportional to its displacement, and these quantities act in the opposite direction. A common example is a mass on a spring.
We can look at the potential energy of the system, which happens to be quadratic for a simple harmonic oscillator. Thus, we can make an analogy for our system by thinking of a ball rolling around in a quadratic potential well. The position of the ball matches the position of the real harmonic oscillator, and the height of the ball inside the well relates to the energy of our harmonic oscillator system.
This quadratic potential can also be translated to a quantum system. In other words, we can study a quantum harmonic oscillator (QHO). QHOs are difficult to find in real life, but many systems approximately behave like QHOs. The biggest difference between a QHO and a classical harmonic oscillator is that the QHO can only be found with specific energy values. The classical one can take any possible value in the range determined by the potential well.
The quantization of the QHO allows us to ask a few important questions. Firstly, what are the actual energy values of each allowed energy level? Secondly, how far apart are these energy levels? And thirdly, can our system transition between different energy levels over time?
This is where ladder operators come in. Without them, we would have to solve the Schrodinger equation (the governing equation of quantum mechanics) for every single energy level in order to find the energy values and the gaps. But it turns out our system can transition from one energy level to another if energy is supplied externally (such as by a photon) in order to go up in energy. And energy is released if the transition is downward.
Instead of solving the Schrodinger equation multiple times though, we can just solve it once, and then apply ladder operators to the found solution. The creation operator calculates the wave function of the most adjacent higher energy level, given the wave function we calculated from the Schrodinger equation. The annihilation operator goes the other way, calculating the wave function of the level just below. Thus, we can find all the wave functions by simply applying the ladder operators to one solution.
Additionally, the wave function allows us to count the number of quanta (e.g. photons) needed to get to a particular state from the lowest (ground) state. And with a restatement of the Schrodinger equation, we can also find a simple expression for the energy of a particular energy level without having to solve the Schrodinger equation!
Hence, the ladder operators are really as interesting as they sound, just not in the way we'd necessarily expect upon first hearing their names!
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Timestamps:
0:00 - Creation and Annihilation Operators (Ladder Operators)
0:36 - The basics: Classical Harmonic Oscillator (Mass on a Spring)
1:46 - Potential Energy (and the Quadratic Potential Well)
3:01 - The Quantum Harmonic Oscillator, and Quantization
5:19 - A Brief Overview of Quantum Wave Functions
5:45 - Questions About the Quantum Harmonic Oscillator
6:26 - The Ladder Operators vs the Schrodinger Equation
7:01 - The Creation Operator
7:46 - The Annihilation Operator
8:01 - Why They're Called Ladder Operators
8:33 - Benefits of the Ladder Operators
9:18 - Counting the Number of Quanta in Our System
10:50 - The Energy of Each State
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2 авг 2024
