Quantum Chemistry: Understanding the Schrödinger Equation
Many applications of computational chemistry involve solving the Schrödinger equation, but what does that mean and why do we care? We need to understand this equation before we can think about how to apply quantum computing so here I’m breaking down some of the terminology so we can understand how to apply it.
The Schrödinger equation is a partial differential equation that describes a quantum system. The following is the time dependent Schrödinger equation. It looks terrifying now, but we are going to break it down and talk about what it means to solve it.
Lets start with something a little less frightening. The Schrödinger equation in its simplest form is an eigenequation. I covered the definition of an eigenequation in a earlier post and so if you are not familiar with the concept then I recommend ducking out for a moment then returning once you have read that post.
In this equation |𝛹⟩ is the wave function which describes the quantum state of the system. H-hat is the Hamiltonian operator that describes the energy of the particles of the system and their interactions and we’ll come back to that later. E is the total energy of the system. A solution to the Schrödinger equation for any given Hamiltonian H, is a wave function that is an eigenfunction of the Hamiltonian.
Why do we want to find solutions to the Schrödinger equation?
Solutions to the Schrödinger equation represent stationary states of the quantum system and the corresponding eivenvalues represent allowable energy states of the system. Therefore when you solve the Schrödinger equation you find the allowable energy states of the system.
Hold onto that thought. It is going to be important when we start to look at applying quantum computing to computational chemistry in later articles.
One cool property of the Schrödinger equation is that any linear sum of solutions is also a solution to the Schrödinger equation. So the wave function can be linear sum (or superposition) of eigenstates, which starts to look a bit like the quantum wave form expressions we have seen before:
|ψ⟩ = α |ψₐ⟩ + β |ψᵦ⟩
When the system is in a superposition of eigenstates, instead of observing one eigenvalue with certainty, you will instead observe various eigenvalues with various probabilities. That is of course until you measure the system and collapse the wave function.
Help I’ve got lost!
If all this talk of eigenfunctions has got you in a twist then take a breath and don’t panic. What I want you to take away from here is that applying the Schrödinger equation involves defining an operator called a Hamiltonian then finding a wave function or set of wave functions that fit the mathematical properties above relating to that Hamiltonian operator. Those functions describe the stable states of the system and each of those states is associated with a value we can measure for a certain observable of the system, such as energy, spin or polarity.
But what is the Hamiltonian operator?
The Hamiltonian operator is a description of our quantum system and a linear operator on the space of wave functions. It comes in two parts: the kinetic energy of the system and the potential energy of the system. The kinetic energy, T describes the energy of each of the particles in our system independently. The potential energy, V describes the energy of the interactions in the system.
The kinetic and potential energy operators can be expressed as above, where p-hat is the momentum operator, m is mass, ℏ is the reduced Planck constant and r is position. The momentum operator is a differential operator and so the dot product of the momentum operator makes the Hamiltonian a second order differential operator. The upside down triangle is the del operator, which represents the sum of partial differentials across a number of dimensions. The dot product of the del operator is called the Laplacian and in three dimensions looks like this:
The complexity of the Hamiltonian increases as we understand more about the quantum universe and add more terms to the expression, but the differential term in the kinetic energy is not more complicated than the differential operator we met when introducing eigenfuctions. The Hamiltonian operator is a transformation on the wave function and the differential operator term just returns the partial derivative of the wave function.
What do we do with the Schrödinger equation now?
So to apply the Schrödinger equation you first build the Hamiltonian for your system and then you solve the equation to find the eigenstates of the Hamiltonian and to build the waveform. Unfortunately, even a simple Hamiltonian to model the energy states of electrons on a hydrogen atom gets complicated as we add in terms for the location, size, shape and fluctuations of the nucleus and we include the effect each electron has on each other. Then you need to add in some effects from quantum field theory and relativity and scale up to not just one atom, but to large complex molecules that are interacting. You can imagine that as soon as you have larger atoms or whole molecules, the Hamiltonian gets very big and very complex very quickly.
What about time?
So far we have looked at the time-independent form of the equation. The equation can be further generalized to take into account time:
Breaking that equation down, we have the wave function relative to time t, |𝛹(t)⟩. We also have the i, which is the square root of -1, and ℏ, the reduced Planck constant, and we also have the Hamiltonian operator denoted by H-hat. Solving this equation is more complex, but given H, solving the equation for the time-independent case gets us some of the way.
Applying the Schrödinger equation to a real problem
To apply the Schrödinger equation for a quantum system we first build the Hamiltonian from what we know about the particles in the system and the interactions between them. We then solve the Schrödinger equation for that Hamiltonian. This means finding the eigenfunctions of the Hamiltonian, which define the waveform and the stable states of the system such as allowable energy levels or likely position of an electron. These properties then tell us how the system will behave, which is the goal of computational chemistry.
The image below shows the probability density plots for hydrogen atom electrons as derived from the Schrödinger equation. Each plot is an eigenstate for the Hamiltonian and a solution to the Schrödinger equation. I won’t go into how you do this here, but it highlights what you can do with this very powerful bit of linear algebra.
