What Is An Eigenfunction / Eigenvalue Problem Anyway? | Fundamentals of Quantum Mechanics

Quantum Chemistry Problem [Q20-00-00]
Fundamentals of Quantum Mechanics: What is an Eigenfunction / Eigenvalue Problem Anyway?
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This is a video in our "Fundamentals of Quantum Mechanics" series. In today's video, we will talk about Eigenfunction / Eigenvalue problems in Quantum Mechanics. If you have watched some of the other videos on our channel, you know our focus is on solving chemistry problems. But this video will be a little different, because instead of working out a problem, this video will try to answer a simple question: "What is an eigenfunction / eigenvalue problem anyway?"

[*] A key equation in quantum mechanics is the Schrodinger's equation, or more precisely, the time-independent Schrodinger Equation: H Psi = E Psi. In mathematics, this equation is called an eigenfunction / eigenvalue problem. Let's take a closer look at what it means.

Let's say our system is one-dimensional. [*] In this case, the "wavefunction" Psi is a function of x. Representing this explicitly, we have: H Psi(x) = E Psi(x).

[*] On the left side, H is an "operator". [*] An operator "operates" on the function following it, transforming Psi(x) to some output function -- [*] say f(x). The right side states that the output f(x) is just equal to the original Psi(x) [*] multiplied by a number E, not including zero. [*] To distinguish between operators and numbers, we sometimes put a special upside-down v symbol above H.

[*] So what are some examples of "operators"? [*] Here is one: d/dx. d/dx when applied to the [*] purple Psi(x) function morphs it into [*] the green f(x) function. In this example, Psi(x) is a cosine function, so f(x) is a sine. [*] Even though Psi and f are both periodic, the purple Psi(x) lags the green f(x) by a quarter of a period, [*] so there is no way to multiply Psi by just a number and make it come out equal to f.

So we see, in general, some operator, operated on some function Psi, will usually NOT give back the same function.

So what does it mean to "solve" the Schrodinger equation? [*] It turns out that for each operator H that satisfies a certain mathematical requirement, called "Hermitian", you can always find a special set of functions Psi that when operated on by H would give you back the same function, multiplied by a number. [*] These special functions are called "eigenfunctions" of the operator H. [*] The number E corresponding to each eigenfunction is called its "eigenvalue".

[*] Let's look at an example. [*] It turns out that the operature d square dx squared is a Hermitian operator. [*] The cosine functions are eigenfunctions of d square dx square. This is clear since the derivative of cosine is negative sine, and the derivation of negative sine is negative cosine. [*] This is shown as the green function f(x). [*] We see that we can simply multiple Psi by -2 and turn it into f(x). So the purple Psi function is an eigenfunction of d square dx square, [*] and corresponding eigenvalue, E, is -2. [*] So the sine funciton solves this eigenfunction / eigenvalue equation. In fact, you can easily see that not only this, but all sine functions are eigenfunctions of d square dx squared.

Let's look at another example. [*] The operator H being negative d square dx squared + one-half x squared is also Hermitian. The graph shows the first three eigenfunctions of this H. [*] There are actually an infinite set of eigenfunctions of this operator.

To illustrate that these are eigenfuctions, we can take the first one, [*] Psi1 and calculate one-half x squared times Psi. This is shown in yellow [*]. We can also calculate minus d square Psi dx squared. This is shown in red [*]. And if we add the yellow and red curves, we get back Psi1 [*]. In quantum chemistry, this H operator is associated with molecular vibrations and is called the "harmonic oscillator". [pause for animation to complete]

In this next example, we return to the operator we looked at earlier: [*] minus d square dx squared, but this time, we will restrict the solution to be zero everywhere, outside of x = plus/minus .5. This requirement is represented by the [*] orange shaded regions in the graph. The eigenfunctions must be zero in these "forbidden" regions. The eigenfunctions are, again, sinusoidal functions. But, now to satisfy the special boundary condition, they are cosine functions of half-integer periods, i.e. 1/2, 3/2, 5/2 periods, etc., that fit within the allowed region, and at the same time, sine functions of integer periods, i.e. 1, 2, 3 periods, etc., that fit within the allowed region. [*] Again, there are an infinite number of eigenfunctions for this eigenfunction / eigenvalue problem that also meet the special boundary conditions here. In quantum chemistry, this is called "the particle in a box" problem, and is the simplest example of quantum confinement.

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[rebeccastorck]

Very good video!!! I’m surprised you only had 999 subscribers! Now you have 1000. Thank you for the video and explanation