Introduction to Introduction to Quantum Mechanics

Quantum mechanics is a heavy subject in many aspects, but it is one of the most important theoretical frameworks that shapes chemistry. One would be lying, to others or oneself, that it is possible to understand even 10% of quantum mechanics in one semester. We will begin with the basics of quantum mechanics, and the stories (with some physics and math) leading to it. Most of the information in these notes is adapted from Principles of Modern Chemistry by Oxtoby.

Quantization

The idea of quantization came from an experimental measurement. German physicist Max Planck was puzzled by the interaction of solid objects with radiant energy, known as blackbody radiation. You might have already witnessed or have an idea of this phenomenon. Objects emit energy from their surface in the form of radiation, and such energy is carried by electromagnetic waves. The specifics of the wave depend on the temperature of the object. At usual temperatures the radiation falls inside the infrared region, and hence we see pictures of “night vision” of people using infrared detectors. The puzzling observation of blackbody radiation showed up when objects are at high temperatures.

Physics before quantum mechanics predicted blackbody radiation is a result of oscillating electrical charges that are accelerated by thermal motion at the surfaces of objects. Each motion is accompanied by a period, which is inversely proportional to the frequency. Disregarding the details, we will quote the result of what classical physics predicted of blackbody radiation as

$$\rho_T(\nu) = \frac{8\pi k_B T \nu^2}{c^3}$$

where $\rho_T(\nu)$ is the intensity of the radiation at frequency $\nu$. However, this prediction is almost completely wrong at high frequencies (low wavelengths)! Classical physics describes the energy of an oscillating motion to be continuous, taking any real number value, and the consequence of the failure was that scientists had to rethink what physics was. To explain the “weirdness”, Planck hypothesized that the energies of the oscillating motions must not be continuous, but instead discrete. Specifically, each motion either gains or loses energy in the unit of a quanta, with magnitude $h\nu$, where $h$ is what we call Planck’s constant with units of energy × time. Let’s suppose we do not know the value yet. The total energy of an oscillating motion can then be described as

$$\varepsilon_\text{osc} = nh\nu$$

where $n = 1, 2, 3, 4\ldots$

Now suppose each oscillating motion is a quantized oscillator. On the surface of an object there are numerous of these oscillators, each with energies described by the equation above. Planck derived that under the quantized formulation, the new relationship between intensity of radiation and frequency is

$$\rho_T(\nu) = \frac{8\pi h \nu^3}{c^3}\frac{1}{e^{h\nu/k_BT} - 1}$$

At the moment, you might be inclined to ask the origin of the fraction $\frac{1}{e^{h\nu/k_BT} - 1}$; unfortunately, the answer to that is much out of the scope of the class, and I ask you to accept its presence for now. The curious student can refer to online resources on the Boltzmann distribution and partition function, though the details are at sophomore/junior level chemistry.

Remember we held off the value of Planck’s constant $h$ earlier? Now we could think of its numerical value. Given the newly derived distribution of radiation intensity, we can determine the best value of $h$ such that the plotted values fit best with the experimental result. Over the years, physicists have determined the value of $h$ using many other techniques, both experimentally and theoretically. The current accepted value is

$$h = 6.62606957 \times 10^{-34}\ \text{J s}$$

Let’s now summarize the main ideas and consequences of blackbody radiation, particularly how Planck’s hypothesis (which turns out to be correct and fundamental for quantum mechanics) overthrew what people thought of classical physics.

1. The energy of a system can take only discrete values, which are represented on its energy-level diagram.
2. A quantized oscillator can gain or lose energy only in discrete amounts $\Delta\varepsilon$, which are related to its frequency by $\Delta\varepsilon = h\nu$.
3. To emit energy from higher energy states, the temperature of a quantized system must be sufficiently high to excite those states.

These three ideas are the basis for our understanding that energy is discrete, not continuous, and that it can be transferred only in discrete chunks and not by arbitrary amounts. Every system has its own energy-level diagram that describes the allowed energy values and the possible values of energy transfers.

Exercises

1. Verify that Planck’s distribution reduces to what classical physics predicts at high temperatures, i.e. $T \to \infty$. Hint. Recall the Taylor expansion $e^x \approx 1+x$ for small $x$.

2. An FM radio station broadcasts at a frequency of $9.86\times 10^7\ \text{s}^{-1}$ (98.6 MHz). Calculate the wavelength of the radio waves.

3. The speed of sound in dry air at 20 °C is 343.5 m/s, and the frequency of the sound from the middle C note on a piano is 261.6 s⁻¹ (according to the American standard pitch scale). Calculate the wavelength of this sound and the time it will take to travel 30.0 m across a concert hall.

4. The radius of an atom of gold (Au) is about 1.35 Å. (a) Express this distance in nanometers and picometers. (b) How many gold atoms would have to be lined up to span 1.0 mm? (c) If the atom is assumed to be a sphere, what is the volume in cm³ of a single Au atom?

5. Only two isotopes of copper occur naturally, $^{63}$Cu (atomic mass 62.9296 amu, abundance 69.17%) and $^{65}$Cu (atomic mass 64.9278 amu, abundance 30.83%). Calculate the atomic weight of copper.

Schrödinger Equation

Schrödinger, an Austrian physicist, reasoned that an electron (or any other particle with wavelike properties) might well be described by a wavefunction. What is a wavefunction exactly? One could understand it from many perspectives. The most general description is that it is a mathematical function in a mathematical space with specific mathematical properties that contains all the information about your desired physical system, e.g. an electron. It is important to note that the wavefunction itself has no physical meaning. Only the absolute value squared of the wavefunction has a physical meaning, the probability distribution of the location of your physical system. Just like in classical physics there is an equation of motion, the famous $F = ma = m\frac{d^2x}{dt^2}$, the theory of quantum mechanics needed an equation to describe the motion of the wavefunction.

The task of deriving the Schrödinger equation is incredibly daunting, and to many chemists it is generally accepted as the correct equation. However, we will try to go through an intuition behind the Schrödinger equation and work through the logic behind it. The classical wave equation, $\frac{\partial^2 f}{\partial t^2} \propto \frac{\partial^2 f}{\partial x^2}$, relates the second derivatives of a wave with respect to position to the second derivatives with respect to time. It is therefore natural to consider the derivative of the wavefunction in quantum mechanics as well. For simplicity we will omit the time variable, and see if we can find a wave equation that relates the second derivative of a function with respect to position to the function itself. (The curious student might ask why. While out of scope of the class, keep in mind that quantum mechanics is almost all built around linear algebra and eigenvalue problems.)

We begin by considering a particle moving freely in one dimension (along the $x$-axis) with classical momentum $p$. Such a particle is associated with a wave of wavelength $\lambda = h/p$. Suppose I give you the solution to this system, that is, two “wave functions” that describe such a wave are

$$\psi(x) = A \sin\!\Big(\frac{2\pi x}{\lambda}\Big) \;,\quad \psi(x) = B \cos\!\Big(\frac{2\pi x}{\lambda}\Big)$$

where $A$ and $B$ are constants. Without loss of generality, let’s inspect the sine function. Its first derivative is

$$\frac{d\psi(x)}{dx} = A\frac{2\pi}{\lambda} \cos\!\Big(\frac{2\pi x}{\lambda}\Big)$$

and its second derivative is

$$\frac{d^2\psi(x)}{dx^2} = -A\Big(\frac{2\pi}{\lambda}\Big)^2 \sin\!\Big(\frac{2\pi x}{\lambda}\Big)$$

Recall our wavefunction is $\psi(x) = A\sin(2\pi x/\lambda)$, so the second derivative can be written as

$$\frac{d^2\psi(x)}{dx^2} = -\Big(\frac{2\pi}{\lambda}\Big)^2 \psi(x)$$

This is a differential equation, and the solution is given by $\psi(x)$. One can also verify that the cosine function satisfies the same differential equation, hence another solution. Recall the de Broglie relation $\lambda = h/p$ and substitute it in. We now get

$$\frac{d^2\psi(x)}{dx^2} = -\Big(\frac{2\pi}{h}p\Big)^2 \psi(x)$$

We can rearrange the equation by multiplying $-\frac{h^2}{8\pi^2 m}$ on both sides, giving us

$$-\frac{h^2}{8\pi^2 m}\frac{d^2\psi(x)}{dx^2} = \frac{p^2}{2m}\psi(x) = \mathcal{T}\psi(x)$$

where $\mathcal{T}$ is $\frac{p^2}{2m}$, the kinetic energy. Recall our setup is a particle moving in free space, which means it only has kinetic energy. When we include external forces, such that there is an external potential energy $V$, we write the equation as

$$-\frac{h^2}{8\pi^2 m}\frac{d^2\psi(x)}{dx^2} = \frac{p^2}{2m}\psi(x) + V(x)\psi(x)$$

which we call the Schrödinger equation. When you google the Schrödinger equation you will most likely encounter different forms of it, due to pedagogy naming discrepancies that evolved through the years. The equation shown here is the time-independent Schrödinger equation. Generally, the Schrödinger equation implies a dependence on time, which is out of the scope of this class.

And that’s it! Please contact me if you have confusion or discovered any mistakes.