The Weapon of Quantum Mechanics
Day 4 - Quantum30 challenge
Welcome to Day 4 of QuantumComputingIndia’s Quantum30 challenge! Today, we delve into the heart of quantum mechanics to explore the fundamental concepts of wavefunction and the Schrödinger equation.
(Side note: The quoted sentences imply a postulate of Quantum Mechanics. I have not mentioned all of them but only the relevant ones for now.)
The Quantum Trishul
Our Quantum weapon looks very similar to this. It is a mathematical expression denoted by the Greek letter ‘Psi’ (𝝍) and this letter, in general, represents a wavefunction.
The state of a Quantum System is completely described by the wavefunction which is a function of position and time
The wave function is a function of position and time and encompasses all the vital information about a Quantum particle like position, momentum, energy, angular momentum, etc. These pieces of information are called ‘observables’ as these quantities can be, well, observed or measured. But here’s a catch! We cannot easily get all this information. In order to know about any observable, we have to feed the wavefunction with something. This ‘something’ is called an Operator.
Every observable has a corresponding operator
So when one wants to know the kinetic energy of the system, we use the corresponding operator on the wavefunction, and Voila! We get to know about the KE of the particle.
‘THE’ Equation
The wavefunction evolves according to the Schrödinger equation
Welcome back Shrodinger!
The wave function is not a static entity; it evolves with time and space. The evolution of the wavefunction is governed by the Schrödinger equation, which is the cornerstone of quantum mechanics. There are two forms of the equation viz. the time-dependent and time-independent
This looks intimidating at first, but understanding it can calm us down a bit. In both of the equations, the first term is the kinetic energy operator operating on 𝝍 whereas V represents the potential energy operator operating on 𝝍. The right-hand side of the equation is the total energy or as we like to call it in Quantum Physics the ‘Hamiltonian’ of the system. So overall, the Schrodinger equation is nothing but an equation of energy and in a simplified way, we can say that this equation is TE = KE + PE.
Unraveling Quantum Behavior: The Trapped Particle Example
To illustrate the power of the Schrödinger equation, let’s consider a ‘classiq’ example
Suppose, we have a Quantum particle trapped in a box. Within the box, the particle can move freely but is confined by an infinite potential at the boundaries, preventing it from escaping. We are also given the dimensions of the trap and now, all we have to do is solve the Schrodinger equation.
By solving, we get the equation of the wavefunction of a particle trapped and the final step is operating to get information about an observable.
As pointed out many times before, Quantum mechanics deals with probability or the probability amplitude which is nothing but the absolute value square of the wavefunction in space. By squaring the wave function, we get the probability of where the particle can be in the system.
Quantization of Energy
The Schrödinger equation also sheds light on why energy is quantized in quantum systems. Again going back to the trapped particle, as the potential at the boundaries is zero, this means, the wavefunction at that point should be zero i.e. a node is formed at the ends.
In a classical picture, this is very similar to a string stretched and clamped at both ends. When we try to vibrate it, only certain types of vibrations are possible and hence we can only get certain frequencies.
Similarly, our trapped particle can execute only certain types of vibrations. And as energy and frequency are related by the equation E = hf (h is Planck's constant), we can see that there will be only certain energy values possible.
Conclusion
The wavefunction and the Schrödinger equation are essential tools in quantum mechanics, empowering us to comprehend and predict the behavior of quantum systems. While this article offers a glimpse into these intriguing concepts, a comprehensive understanding requires further study and exploration of the mathematical formalism that underpins the quantum world. Happy quantum exploring!