QM II: Why Quantum Mechanics?

1.Why Quantum Mechanics?

1.1.A Brief History

In last term's lectures, you have explored a powerful reformulation of ‘classical mechanics’ using the Lagrangian and Hamiltonian formalisms. This is perfectly adequate for a complete understanding of balls, pendulums, springs, and waves, on macroscopic distance scales familiar to humans.

However, by the early 1900's there was mounting experimental evidence that the elementary constituents of matter, at microscopic distances of atoms and molecules, behave in a wholly different manner. The new theoretical framework that emerged in this period to describe such phenomena is known as ‘quantum mechanics’. This has had a profound effect on society: quantum mechanics underpins much of the technological revolution of the last century.

Ultimately, the framework of quantum mechanics is determined by experimental facts. However, many features of quantum mechanics are reminiscent of the Hamiltonian formulation of classical mechanics, and we will emphasise this connection throughout the course. You should keep in mind that quantum mechanics is the more fundamental description of nature, with classical mechanics an approximation valid at macroscopic distances.

Introductory lecture, in which the module is introduced and motivation is given that led to the birth of quantum mechanics.

Quantum mechanics has an extremely rich mathematical framework. In this course, we will encounter techniques from analysis, probability, algebra, and representation theory. Moreover, ideas and techniques from quantum mechanics have inspired many exciting developments in pure mathematics in the last half century, particularly in geometry and topology. This continues to be an active area of research today.

In this lecture and the next, we will explore some of the inconsistencies between classical mechanics and experimental facts about nature at microscopic distances. This will serve as a guide to the development of quantum mechanics in subsequent lectures.

1.2.Bound States

Many potentials which lead to a continuum of bound states in classical mechanics produce, instead, a discrete spectrum in the real world, which can be ‘explained' by `quantisation conditions’.

In classical mechanics, a particle has a definite position and momentum $(x(t), p(t))$ at each time $t$. We can view this geometrically as a curve in ‘phase space’ parametrised by time $t$. Given some initial conditions, the shape of the curve is determined by Hamilton's equations,

\begin{equation} \label{firsteq} \dot x = \frac{\partial H}{\partial p}\,, \qquad \dot p = - \frac{\partial H}{\partial x} \, , \end{equation}

where $H$ is the Hamiltonian. More precisely, Hamilton's equations specify the tangent space to the curve at each time $t$, as shown below.

Figure 1.1: Phase space dynamics of a classical system, driven by the Hamiltonian.

Example. A particle of mass $m$ moving in a potential $V(x)$ has
\begin{equation} H = \frac{p^2}{2m} + V(x) \, . \end{equation}
The Hamiltonian itself is conserved and equal to the total energy $E$, given by the sum of kinetic energy and potential energy. In this case, eliminating the momentum $p$ from Hamilton's equations leads back to Newton's law,
\begin{equation} m \ddot x = - \frac{dV}{dx} \, . \end{equation}

In classical mechanics, a ‘bound state' is a solution of Hamilton’s equations that is confined to a finite region of phase space. Bound states arise from oscillations around a local minimum of the potential $V(x)$.

Example 1 A simple harmonic oscillator has quadratic potential
\begin{equation} V(x) = \frac{1}{2}m\omega^2x^2 \, , \end{equation}
where $\omega$ is known as the ‘angular frequency’. The general solution of Hamilton's equations is given by
\begin{equation} \label{e:harmonic_oscillator_solution} \begin{aligned} x(t) & = \sqrt{\frac{2E}{m\omega^2}} \sin(\omega t + \phi )\,,\\[1ex] p(t) & = \sqrt{2mE} \cos(\omega t + \phi ) \, , \end{aligned} \end{equation}
where the energy $E \geq 0$ and phase $\phi$ are determined by the initial conditions.

Figure 1.2: Potential energy and phase space dynamics of the simple harmonic oscillator.

The particle is confined to the region where $V(x) \leq E$ and cannot escape to infinity. The curve $(x(t),p(t))$ forms an ellipse in phase space, which is confined to a finite region. There is therefore a continuous spectrum of bound states parametrised by the energy $E \geq 0$.
Example 2. The effective potential for the radial motion of an electron in a hydrogen atom is (in convenient units)
\begin{equation} V(x) = \frac{J^2}{2mx^2}-\frac{e^2}{x} \, , \end{equation}
where $m$, $e$ are the mass and electric charge of the electron and $J^2$ is the conserved square of the angular momentum vector. The potential has a minimum at $x_{0} = J^2/me^2$ and asymptotes to $0$ from below as $x \to \infty$. There is a continuous spectrum of bound states parametrised by $J^2$ and energy $V(x_0) \lt{} E \lt{} 0$.

Figure 1.3: Classical potential energy of the hydrogen atom; there is a classical bound state for every energy $V(x_0)\lt{}E\lt{}0$.

More generally, any potential $V(x)$ with a local minimum at some point $x_0$ will classically have a continuous spectrum of bound states with energy $V(x_0) \lt{} E \lt{} E_{\mathrm{max}}$ for some maximum energy $E_{\mathrm{max}}$.

This classical expectation is in striking contradiction with experimental tests of microscopic systems, which typically have a discrete spectrum of bound states. For example, from the study of atomic spectra it is known that a hydrogen atom has a discrete set of bound states where angular momentum and energy take particular values

\begin{equation} \begin{aligned} J^2 & = j(j+1)(2\pi h)^2 \qquad & j &= 0,1,\ldots \,,\\[1ex] E & = - \frac{(2\pi)^2 me^4}{2 h^2 n^2} \qquad & n & = 1,2,\ldots \, . \end{aligned} \end{equation}

The quantity $h$ is a new constant of nature known as the Planck constant. This has units of 'energy times time' (just like angular momentum) and is approximately

\begin{equation} h \approx 6.63 \times 10^{-34} \, \mathrm{kg \, m^2 \, s^{-1}} \, . \end{equation}

The notation $\hbar = h/(2\pi)$ is also commonly used, and called the reduced Planck constant,

\begin{equation} \hbar \approx 1.05 \times 10^{-34} \, \mathrm{kg \, m^2 \, s^{-1}} \, . \end{equation}

These discrete spectra of bound states cannot be explained in the framework of classical mechanics. Later in the course, we will show that a discrete spectrum of bound states is, however, a characteristic feature of quantum mechanics.

In the beginning days of quantum mechanics, people tried to impose these quantisation rules by hand, for instance by imposing that for the harmonic oscillator,

\begin{equation} \label{e:one} \int_{\text{orbit}} p {\rm d x} = n h\,, \quad n\in{\mathbb Z}\,, \end{equation}

where the integral is over one entire orbit. For the harmonic oscillator, we get (see the problems)

\begin{equation} \int_{\text{orbit}} p {\rm d x} = \frac{2\pi E}{\omega}\,, \end{equation}

and so with $\omega = 2\pi\nu$, the `quantisation condition' \eqref{e:one} produces $E = \nu h n$. However, such quantisation prescriptions clearly do not constitute a ‘theory’, and are difficult to generalise to more complicated systems.

1.3.The Photoelectric Effect

Light irradiated onto a metal behaves as if it is made of discrete constituents (‘photons’).

We now consider another important phenomenon that is inconsistent with classical mechanics. The `photo-electric effect' is the emission of electrons from certain metals when irradiated by light.

Figure 1.4: Schematic depiction of the photo-electric effect: when light of frequency $\omega$ is irradiated onto a metal, it releases electrons.

In classical mechanics, light is a wave. This is a fluctuation in the electromagnetic field that solves the wave equation

\begin{equation} \frac{\partial^2 \psi}{\partial t^2} - c^2 \frac{\partial^2 \psi}{\partial x^2} = 0 \, , \end{equation}

where $c$ is the speed of light. You can think about the real part of the amplitude $\psi(x,t)$ as a component of the electric or magnetic field. Let us assume the light is monochromatic and accurately described by a plane wave

\begin{equation} \psi(x,t) = \psi_0 \, e^{i \omega (x/c-t)} \end{equation}

with angular frequency $\omega$. This is related to the wavelength by $\lambda = 2\pi / \omega$. The ‘intensity’ of the light is the energy carried by the electromagnetic field, averaged over time. This is proportional to the modulus squared of the amplitude, $I \sim |\psi(x,t)|^2 = |\psi_0|^2$, and is independent of the angular frequency $\omega$.

Let us assume an electron in the metal must absorb a minimum amount of energy $E_{\text{min}}$ from the light to be emitted from the metal. Then the classical description of light as a wave leads to the following expectation:

The energy of the emitted electrons depends on the intensity $I$ but is independent of the angular frequency $\omega$.
Electrons are emitted even in low-intensity light, but there is a time-delay as each electron absorbs the minimum energy $E_{\text{min}}$.

However, the experimental result is the following:

The energy of emitted electrons is independent of the intensity $I$ and is linearly proportional to the angular frequency,
$\hbar \omega - E_{\text{min}}$.
Electrons are only emitted if $\hbar \omega \geq E_{\text{min}} $ and are emitted immediately.

Here $\hbar$ is the same Planck's constant introduced above. This is shockingly different to the classical expectation!

In 1905, Einstein made a remarkable proposal that resolved this contradiction: that light arrives in indivisible packets known as ‘quanta' or `photons’. The energy carried by each individual photon is

\begin{equation} E = \hbar \omega \, , \end{equation}

while the intensity is related to the rate that photons are arriving. Assuming an electron can only absorb one photon at a time, this means that an electron can only be emitted if $\hbar \omega \geq E_{\text{min}}$. Its energy is equal to that of the photon it absorbs minus the energy needed to escape the metal, $\hbar \omega - E_{\text{min}}$.

1.4.These lectures

In these lectures we will restrict to one-dimensional quantum mechanics in the Schrödinger wave function approach.

We have seen two examples of how the classical mechanics of particles and waves fails to explain experimental data at microscopic distances. Furthermore, we have seen hints that light has characteristics of a wave, but arrives in indivisible packets like a particle. This is known as `particle-wave duality' and is a feature not only of light but also electrons and all constituents of matter.

In the following chapters, we will explore this idea much more precisely, starting with the ‘double slit’ experiment, and then gradually developing a mathematical formalism that can explain the phenomena discussed above.

Feynman on the difficulty of understanding quantum mechanics.

As we go along, you will discover that quantum mechanics is a hard topic. This is not only because it requires you to understand a load of new mathematical ingredients, but mostly because, as a beginner, you will be guaranteed to lack an intuition for it. Your classical experience with the real world out there is of no use when it comes to understanding the microscopic world governed by quantum mechanics. So the only way to ‘gain intuition’ is to solve many problems and slowly get used to the strange miscroscopic world. To make life easier, we will stick exclusively to one-dimensional systems in these lectures.

Another aspect which does not make it simpler for a newcomer to grasp the concepts is that there exist three different mathematical formulations of quantum mechanics, all equivalent in sofar this can be verified, but radically different in their notation and even conceptual interpretation. The present notes follow the so-called Schrödinger wave function approach, which connects most clearly to classical wave mechanics which was at the root of the development of quantum mechanics originally.

The first of the other two approaches is the operator approach, which formalises much of the wave function approach into the language of operators acting on infinite-dimensional vector spaces. We will touch briefly on this towards the end of the module. Finally, there is the path integral approach, which is both conceptually and technically entirely different from the first two. These two other approaches will be discussed in the Quantum Mechanics III module.

1.5.Recommended literature

Read books! No honestly, read books! There is no substitute for solving problems yourself, but the next best thing is to read multiple sources so you get to see things from different angles.

Three books which these notes to a large extent based on are

Introduction to Quantum Mechanics, David J. Griffiths
A standard textbook. Chapters 1-3 cover the same material as this course but in a different order. Overall, this is the most appropriate textbook.
Quantum Physics, S. Gasiorowicz
Chapters 3-7 contain lots of worked examples relevant for this course.
Principles of Quantum Mechanics, R. Shankar
A popular favourite. Chapters 3-7 cover similar material to this course but at a more advanced level. A good investment for ambitious students who wish to progress onto Quantum Mechanics III.

There are various other books which do get referred to frequently, but which are somewhat further away from the present course, e.g.:

Modern Quantum Mechanics, J. J. Sakurai
Another standard book, with a nice motivation for quantum mechanics at the beginning. It does, however, use the operator approach almost exclusively.
Feynman Lectures, Volume III, R.P. Feynmann, R.B. Leighton, M. Sands
A classic everyone should have read. Covers a lot more than we will cover in this module, but the first few chapters are worth having a look at. Available for free online at https://www.feynmanlectures.caltech.edu.
Notes on Quantum Mechanics, D.V. Schroeder
A very new book by an extremely good educator. Has a lot of emphasis on concrete computations, often using Mathematica. Available for free online at https://physics.weber.edu/schroeder/quantum/.

1.6.Problems

Hamilton's equations
Write down Hamilton's equations for a particle in a harmonic potential,
\begin{equation} V = \frac{1}{2}m \omega^2 x^2\,. \end{equation}
and show that its solutions are as stated.

Solution:
▶
The Hamiltonian and its derivatives with respect to $x$ and $p$ are
\begin{equation} H = \frac{1}{2m}p^2 + \frac{1}{2}m \omega^2 x^2\,,\quad \frac{\partial H}{\partial x} = m \omega^2 x\,,\quad \frac{\partial H}{\partial p} = \frac{p}{m}\,. \end{equation}
Hamilton's equations are thus
\begin{equation} \dot{x}(t) = \frac{p(t)}{m}\,,\quad \dot{p}(t) = -m\omega^2 x(t)\,, \end{equation}
which you can solve in a variety of ways, e.g. by taking the time-derivative of the first and then inserting the second, to get
\begin{equation} \ddot{x}(t) = - \omega^2 x(t)\,, \end{equation}
and observing that this is solved by
\begin{equation} x(t) = C \sin(t\omega + \phi)\,. \end{equation}
The constant $C$ is fixed by inserting both $x(t)$ and $p(t)$ into the Hamiltonian, and noting that the result should be $E$.
1
Quantisation condition
Consider the simple harmonic oscillator. Fill in the missing steps in the computation that takes you from the ad-hoc quantisation condition
\begin{equation} \int_{\text{orbit}} p {\rm d x} = n h\,, \quad n\in{\mathbb Z}\,, \end{equation}
to the conclusion that $E=\nu h n$ with $n\in {\mathbb Z}$.

Solution:
▶
Inserting the solution
\begin{equation} \begin{aligned} x(t) & = \sqrt{\frac{2E}{m\omega^2}} \sin(\omega t + \phi )\,,\\[1ex] p(t) & = \sqrt{2mE} \cos(\omega t + \phi ) \, , \end{aligned} \end{equation}
into the integral gives
\begin{equation} \int_{\text{orbit}} p {\rm d x} = 2 E \int_{\text{orbit}} \cos(\omega t + \phi) {\rm d}\left( \frac{1}{\omega} \sin(\omega t + \phi) \right)\,. \end{equation}
By applying the chain rule we can rewrite the complicated differential ${\rm d}(\ldots)$ in terms of ${\rm d}t$. Moreover, one orbit is given by taking $t$ from $0$ to $2\pi/\omega$. Together, this gives
\begin{equation} = 2E \int_{0}^{2\pi/\omega}\!\!\! \cos^2(\omega t + \phi) {\rm d} t = \frac{2\pi E}{\omega}\,. \end{equation}

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1.	Why Quantum Mechanics?
1.1.	A Brief History
1.2.	Bound States
1.3.	The Photoelectric Effect
1.4.	These lectures
1.5.	Recommended literature
1.6.	Problems
2.	The Double Slit Experiment
3.	Wave function and Probabilities
4.	Momentum and Planck's constant
5.	Schrödinger's Equation
6.	The Hilbert Space
7.	Hermitian Operators
8.	The Spectrum of a Hermitian Operator
9.	Postulates of Quantum Mechanics
10.	Commutators and Uncertainty Principle
11.	Energy Revisited
12.	Stationary states
13.	Case Study: The Free Particle
14.	Two-particle systems
15.	Simple Harmonic Oscillator
16.	The Continuity Equation
17.	Scattering Problems
18.	Tunnelling
19.	Momentum-space Wave function
20.	Ehrenfest's Theorem
21.	Bibliography