## Quantum Computing: What is the Schrodinger Equation anyway?

In this article I want to give an very brief introduction to the Schrodinger Equation and how we can use quantum mechanics in computing to achieve things classical computation methods cannot.

Hopefully, this article will go some way in simplifying what all this “Quantum Computing” malarkey is.

I am going to assume mathematical knowledge in this article, and I am only going to cover each concept briefly. Quantum mechanics and quantum computing are extremely wide and deep areas of study; this article will only cover the hight level topics that will hopefully give you some ideas that you can go away and dive into to get a better understanding.

So, with all that said, lets jump right in.

# What is the Schrodinger Equation?

Simply put the Schrodinger equation is a mathematical equation that allows us to predict the probability of obtaining a value from a quantum mechanical system, if we took a measurement of that quantum mechanical system.

The Schrodinger equation is a partial differential equation and looks like this:

The reason it’s a partial different equation is that we have differentials that depend on two variables, namely position x and time t.

The Schrodinger equation can be used to predict what will happen at time t of any wave function we plug into it. The particular example above has the wave function ψ(x,t) and as this wave function only accepts the values x and t then we say that this is one dimensional as we only have one spatial variable x.

The wave function in quantum mechanics is a function that describes the system of interest and it contains everything there is to know about the system. We plug our wave function into the Schrodinger equation and the solutions tell us the probability of getting a value (Eigen value) of a particular observable.

# Operators

An observable in quantum mechanics is something that we can measure. For example, we can measure the energy values an electron can possess when bound to an atom. In this example the observable is the energy value. To get the energy values we need to operate on our wavefunction by using an operator. Each observable is associated with an operator and the energy operator is called the Hamiltonian operator and is denoted by

The total energy of a system is the sum of it’s kinetic and potential energy. We can see in the Schrodinger equation above that we have the term

This is the potential energy.

In classical mechanics we can write the kinetic energy as

where

is the momentum along in the x direction and m is the mass of the particle.

So, the total energy of a system can be written as

This is the Hamiltonian function

We cannot use this form of energy equation in quantum mechanics however we need to use a quantum mechanical operator instead.

The quantum mechanical operator for momentum is

So our quantum mechanical operator for kinetic energy becomes

As you can see in our Schrodinger equation above we have the term

and this represents the kinetic energy term for our system.

We can now form the quantum mechanical Hamiltonian operator which is

This represents the energy observable which we can apply to our wave function to get values for the energy of our system.

So, we can now rewrite the Schrodinger equation with the Hamiltonian representing the energy part:

# Stationary states

The problem with the above equation is that it is a partial differential equation, and these are more difficult to solve. It turns out we can obtain a form of Schrodinger’s equation that is independent of time by using a method called separation of variables.

We can represent our wave function ψ(x,t) as the product of two functions like so

It turns out by separating the wave function like this we can obtain a time independent form of Schrodinger’s equation:

The solutions to this time independent Schrodinger Equation are called stationary states and when subject to boundary conditions give us the energy levels of a system.

It is important to note that the wave function is probabilistic in nature which in simple terms gives us the probability of finding the observable in a state when we measure it. The time dependent Schrodinger equation itself gives us a way of predicting forward in time how the wave function will evolve. Once a measurement is taken of a particular observable however the wave function collapses to ONE particular state.

What we mean by collapsing to a particular state is that a wave function can be made up of combination of states called a linear combination of states.

# Dirac notation

Dirac notation is a more concise way of representing wave function states in quantum mechanics.

We say that a wave function can be a linear combination of other wave functions and this is represented in Dirac notation as

Here c_i is a complex constant. As we can see a wave function can be just a sum of other wave functions.

Another way to represent states using Dirac notation is by using column vectors.

For instance, a state can be represented by the following notation:

# How can we use this Schrodinger Equation for computation?

So first let’s see how classical computers work and then lets see how using quantum mechanics can improve on it.

All computers obey the basic model of a Turing machine. A Turing machine is an abstract model of computation invented by Alan Turing in 1936. You can look up the details of how a Turing machine works but the general notion is that the machine works on a finite set of states and obeys rules for how to compute things.

Modern computers today implement this computation model by using transistors and logic gates. Transistors are essentially tiny little switches that represent either on or off states.

Computation in classical computers happens by representing states in these transistors so there is ultimately a limit of how many states a computer can represent, and the Turing machine can only operate on each state one at a time.

So, for instance we can represent two possible binary states for a classical computer like so:

That’s two possible states and a classical computer can only operate on these states to do computation one at a time.

Let’s contrast that with a quantum computer. Remember we said that a quantum state is a linear combination of other states?

So, in a quantum computer we can represent ALL possible states in a single state like so

This state could represent all the possible states that a computer needs to compute on. We can now use the Schrodinger equation to collapse the state onto the most probable state from its linear combination. The state we get as a result will be the answer to our computation.

So, you can see that instead of a Turing machine iterating through each binary state one by one, the quantum computer can represent all state and perform the computing in just one step.

# Other uses for quantum computers

Computation is just one aspect of quantum computing. We can also use other properties that emerge from quantum mechanics like entanglement. Entanglement is used in Quantum Key Distribution algorithms which use entanglement to detect whether a cryptographic key has been tampered with by an unintended recipient.

The subject of Quantum Key Distribution however can wait for another day.