Understanding Schrodinger Wave Equation In Quantum Mechanics

In the very first article of this series, I talked about the wavefunction. I explained two most important concepts of quantum mechanic (quantum physics in fact): Wavefunction and measurement. Now I want to go further and take you from quantum physics to quantum mechanics. Today, in the 7th article of Month of Equations series, let us learn about one of the flagship equations of physics, the Schrodinger Wave Equation. But before that, for those who missed the first equation, I will give a brief introduction to the concept of wavefunction.

What is a wavefunction?

Quantum physics was born as a result of the debate on the nature of light. Is light made of particles or is it a wave? Scientists found that it is both. In 1897, J.J. Thomson discovered the first particle, the electron. But then came a twist. The Davisson and Germer experiment showed that electrons, so far thought to be material particles, also have a wave nature associated with them. A wave accompanies a material particle. But there is a difference between the wave nature of a photon and that of an electron. The wave of the former is an electromagnetic wave described in terms of electric and magnetic field while that of the latter has no relation with these fields. The wave quantity of material particles is described by a mathematical function, known as the wavefunction ψ(r,t).

Interpretation of wavefunction

The physical interpretation of wavefunction came from a brilliant quantum physicist: Max Born. The Born interpretation says that the square of the wavefunction over a range represents the probability of finding the particle within that range. Did you notice r and t written with ψ? This means that the wavefunction (and thus the probability of finding the particle) is different at different position and time. I will explain this point with a very simple analogy.

Also Read: Top 10 quotes on quantum mechanics that show how beautiful the subject is.

Wavefunction

Suppose you plan to give a surprise visit to your friend. The probability of your friend doing a particular activity depends on the time when you visit him. Suppose you visit him at 5 in the morning. There is a higher probability, say 80%, of finding him sleeping in the bed. There's say 15% probability of finding him taking his bath and 5% probability of finding him studying. But if you visit him at 1 in the afternoon, these numbers will change (Say, 10% sleeping, 85% studying and 5% taking his bath). So the probability is actually a spread and not a certain quantity (unless you open the door). This is exactly what a wavefunction means in quantum mechanics. The particle can be anywhere within the boundaries of the system and the probability of finding the particle at a particular position is different at different places and time.

The Schrodinger Wave Equation

Schrodinger Wave Equation
The Schrodinger Wave Equation

I want you to revisit Newton's second law, F = dp/dt. Why is this equation so important in classical mechanics? The answer is that it solves for the position of the particle. Once you know the position, you can find everything else: momentum, velocity, energy etc. Hence, Newton's second law is the most fundamental equation of classical physics.

The Schrodinger wave equation is the quantum analogue of the Newton's second law. Just as the latter solves for the position of a particle, the former solves for the wavefunction which is the most important entity in quantum mechanics. Once you know the wavefunction, you can calculate anything.

Meaning of Schrodinger wave equation

There are two forms of this equation: Time independent and time dependent. In this article, we will discuss the former. The meaning of the Schrodinger wave equation is, "The total energy of the particle is the sum of its kinetic and potential energy". Yes! That simple. Look at the above equation carefully. ψ(x) is the wavefunction and V(x) is the potential applied. This can be anything. For the case of an electron in hydrogen atom, it can be the electrostatic potential provided by the positively charged nucleus. E is the energy of the system.

The most compact formulation of quantum mechanics is in the form of matrices and operators. In its most compact form, the Schrodinger wave equation is written as H ψ(r,t) = E ψ(r,t). Here H is the operator known as Hamiltonian of the system (the sum of kinetic and potential energy). An operator acts on the wavefunction and produces energy eigen values. So this equation is actually eigen value equation that produces eigen values E of energy. This operator formulation requires deeper understanding of mathematics and hence I will leave this for another time.

Also Read: The life and work of Erwin Schrodinger

Applications of schrodinger wave equation

The Schrodinger wave equation changed the course of physics. Just like Newton's law in macroscopic world, we now had an equation that tells the dynamics of particles in the quantum world. This was the equation that took us from quantum physics to quantum mechanics. When we solve the time independent Schrodinger wave equation for a system, we find that the energies at the quantum scale are quantized. We get a spectrum of energies. This is exactly what Bohr told in his model. Also, the time dependent form tells helps us to know how a quantum system evolves with time.

Despite the success of this equation, there were a few shortcomings in it. This equation was not for relativistic particles. This was rectified by Paul Dirac in his equation that also led to the prediction of antimatter.

Previous in series: What does the Vlasov Equation really mean?

Author's message

As a high school science student, I didn't like matrices and determinants. I once asked my mathematics professor if these are of any use to me in physics. He politely said that these are your tools. If you are going to pursue physics as your career, mathematics is your tool unlike mine. You will remember these words when you will come across quantum mechanics. He was right. Matrices and determinants really play a huge role in understanding quantum mechanics. They make it beautiful and compact.

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