The Schrödinger Equation: A Mathematical Venture
Ask any quantum enthusiast the essence of quantum mechanics and the first thing they’ll answer would surely be Schrödinger’s Cat, the next? Probably the Schrödinger Equation. But what exactly is this equation and why is it so quintessential that it would be memorized on the tongue of any quantum physicist? That is what we’ll cover in the following couple of articles. (Note: heavy math ahead | ExtraNote: The math ahead requires some bit of math that we have already covered in previous articles, apart from a few other topics like complex numbers and very basic partial differential equations | ExtraExtra Note: The LaTex equations don’t look good in dark mode).
So basically, the Schrödinger Equation is a quantum mechanical extension of the classical wave equation and gives you complete information about the energy of a quantum mechanical system. Why use the wave equation for elementary particles like electrons? The answer is wave-particle duality.
As a result, we start with the classical wave equation for a traveling wave: ψ=A(kx-ωt), where A is the amplitude, k=2π/λ=(angular) wavenumber, x=position, ω=angular velocity, and t=time. (Please note that for editing purposes, I’ll write the math inline in Unicode and also attach a LaTex-version here.)
Some textbooks might use the cosine function instead of the sine function in this equation, which is equivalent to the equation we have used, as we have counted in a relative phase. So, for φ=90°, we get the cosine function instead of the sine function. That said, we can eliminate the relative phase by taking φ=0°.
Another form of this classical wave equation you might encounter in literature is ψ=A exp(i(kx-ωt)), which is just the same equation that we have defined above but rewritten in the form of an exponential from Euler’s formula.
So after we have defined our equation, let’s see what happens when we differentiate ψ with respect to time (t) and position (x):
Now, you might wonder what the good thing about these equations is. But, on a closer look, we arrive at some great conclusions. But first of all, we know that k=2π/λ (λ=wavelength). By de Broglie’s relation, k=2πp/h=p/ℏ (ℏ is the reduced Planck’s constant, and p is the momentum).
Similarly, ω=ℏω/ℏ=E/ℏ (where E is the energy), using the Planck — Einstein relation.
Using these two relations in our differentiated classical wave equation, we get:
Now, we define the quantum mechanical momentum operator P-hat such that this equation holds in P-hat as:
Similarly, for energy, we define the energy operator as:
Now, we know from classical mechanics that Kinetic Energy (E) =p²/2m. Using this relation, we get:
The math on simplification yields the boxed equation. This equation is known as the time-dependant Schrödinger Equation, in one dimension.
Adding more dimensions to this question, we get the general form of the equation:
Completing the Equation
One thing you might observe in the equations given above is that all the starting equations, inherited from classical mechanics, are the ones that correspond to the kinetic energy of a body. And yes! Well, that’s true. We have not yet added the potential energy of the body into these equations. Let’s do it.
We define the potential energy of the quantum system as the potential energy operator, V, and write it mathematically as V(x, y, z, t)ψ(x, y, z, t) in 3 dimensions.
Hence, the complete energy equation is given by:
The Time-independent and Space-Independent Equations
So, we have had covered pretty intense math in the previous section right? This would have been pretty much the end of it if φ had not been dependent upon both space and time.
So, can we do something to separate time and space mathematically? Well, yes, but with an assumption. We assume φ to be separable into time and space-dependent functions — u(r) and T(t).
Henceforth, with the assumption in mind, we get the following equations:
Remember the time-dependent kinetic energy equation we earlier left? We’ll use that expression to get our space-independent equation like we got our time-independent equation:
And this is our Time-Independent Schrödinger Equation!
In this article, we have seen mathematically how classical mechanics yields such an important equation of quantum physics. This approach here is called the Free Particle Approach to the Schrödinger Equation. It is to be noted that though the readers might see a few other forms of the Schrödinger Equation too over the internet, all the approaches can basically be boiled down to or derived from these few (not so) simple equations.
For some intuition into this equation, watch out for the upcoming articles!