Quantum Mechanics

Prerequisites: Canonical Quantum Mechanics. Full path (17 pages)

• Intuitive
• Concrete
• Abstract
• Formulations
• Interpretations
• Roadmaps
• History
• FAQ

Intuitive

Quantum mechanics is the theory of how matter and light behave at the scale of atoms and below. It explains why atoms are stable, why they emit light only at certain colours, how chemical bonds form, how transistors and lasers work, and why some materials conduct electricity while others do not. Classical mechanics fails for all of these.

The central change is in what a "state" of a system is. In classical mechanics a particle has a definite position and velocity at every moment. In quantum mechanics the state assigns to every possible outcome of a measurement a number called an amplitude, and the probability of that outcome is computed from the amplitude. The theory predicts these probabilities exactly but, in general, does not predict the outcome of a single measurement.

Amplitudes add, and they can cancel each other. This is what happens in the double-slit experiment. Electrons are sent one at a time towards a wall with two slits, and each electron arrives at the screen behind it as a single dot. After many electrons, the dots form a pattern of bright and dark stripes, the same pattern that water waves passing through two gaps would produce. Where the amplitudes for going through the left slit and through the right slit cancel, no electrons arrive. If we put a detector at the slits to find out which slit each electron took, the stripes disappear.

Two more features follow from the same rules. Bound systems such as an electron in an atom can only have certain discrete energies, much like a guitar string can only vibrate at certain frequencies. And some pairs of quantities, such as position and momentum, cannot both be sharply defined in the same state; this is the uncertainty principle.

For large objects these effects average out and quantum mechanics reproduces the predictions of classical mechanics.

Concrete

The standard formulation rests on a few rules.

• States. The state of a system is a unit vector $|\psi\rangle$ in a complex Hilbert space $\mathcal{H}$, defined up to an overall phase. For a single particle in one dimension we can use the wave function $\psi(x) = \langle x|\psi\rangle$ with $\int |\psi(x)|^2 dx = 1$.
• Observables. Measurable quantities are represented by Hermitian operators $\hat A$. The possible results of a measurement are the eigenvalues $a_n$ of $\hat A$, $\hat A|a_n\rangle = a_n|a_n\rangle$.
• Born rule. For a nondegenerate discrete spectrum, the probability of finding $a_n$ is $P(a_n) = |\langle a_n|\psi\rangle|^2$. The mean of many measurements is the expectation value $\langle \hat A\rangle = \langle\psi|\hat A|\psi\rangle$. After the measurement the state is $|a_n\rangle$ (projection postulate).
• Dynamics. Between measurements the state obeys the Schrödinger equation $$i\hbar \frac{\partial}{\partial t}|\psi(t)\rangle = \hat H |\psi(t)\rangle,$$ where $\hat H$ is the Hamiltonian, the operator of the total energy.

For a particle of mass $m$ in a potential $V(x)$, $\hat x$ acts by multiplication with $x$, $\hat p = -i\hbar\, \partial/\partial x$, and $$\hat H = \frac{\hat p^2}{2m} + V(\hat x), \qquad [\hat x, \hat p] = i\hbar .$$ The commutator implies the uncertainty relation $\Delta x\, \Delta p \ge \hbar/2$ for every state, where $\Delta x^2 = \langle \hat x^2\rangle - \langle \hat x\rangle^2$.

If $\hat H$ does not depend on time, we can expand in its eigenstates, $\hat H|n\rangle = E_n|n\rangle$. A general solution is $$|\psi(t)\rangle = \sum_n c_n e^{-iE_n t/\hbar}|n\rangle, \qquad c_n = \langle n|\psi(0)\rangle .$$ So the main practical task is to solve the time-independent equation $\hat H|n\rangle = E_n|n\rangle$.

Example: particle in a box. Let $V = 0$ for $0 < x < L$ and $V = \infty$ outside. Inside, $-\frac{\hbar^2}{2m}\psi'' = E\psi$ with $\psi(0) = \psi(L) = 0$. The normalized solutions are $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\frac{n\pi x}{L}, \qquad E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \qquad n = 1,2,3,\dots$$ The energy is quantized because only a discrete set of wavelengths fit into the box, and the lowest energy is not zero. A superposition such as $(\psi_1 + \psi_2)/\sqrt2$ has a probability density $|\psi|^2$ that oscillates in time with angular frequency $(E_2 - E_1)/\hbar$, because the cross term carries the relative phase $e^{-i(E_2-E_1)t/\hbar}$.

Composite systems are described by tensor products, $\mathcal{H}_{AB} = \mathcal{H}_A\otimes\mathcal{H}_B$. States that cannot be written as a product $|\psi_A\rangle\otimes|\psi_B\rangle$, for example $(|{\uparrow}\rangle|{\downarrow}\rangle - |{\downarrow}\rangle|{\uparrow}\rangle)/\sqrt2$ for two spins, are called entangled.

The same predictions can also be computed with the other formulations listed below, for example the path integral or the Wigner function on phase space.

• No-Nonsense Quantum Mechanics by Jakob Schwichtenberg, chapter "Bird's-Eye View of Quantum Mechanics"

Abstract

In the Dirac-von Neumann formulation a quantum system is described by a separable complex Hilbert space $\mathcal{H}$.

States are density operators: positive trace-class operators $\rho$ with $\operatorname{tr}\rho = 1$. They form a convex set whose extreme points are the pure states, the rank one projections $\rho = |\psi\rangle\langle\psi|$, i.e. the points of the projective space $\mathbb{P}(\mathcal{H})$.

Observables are self-adjoint, in general unbounded, operators $A$. By the spectral theorem each has a projection-valued measure $E_A$ on the Borel sets of $\mathbb{R}$ with $A = \int \lambda\, dE_A(\lambda)$. The Born rule says that the probability of finding the value of $A$ in a Borel set $\Delta$ is $$P_\rho(A\in\Delta) = \operatorname{tr}\big(\rho\, E_A(\Delta)\big).$$ Gleason's theorem shows that for $\dim\mathcal{H}\ge 3$ every countably additive probability assignment on the lattice of closed subspaces has this form for some density operator $\rho$. Generalized measurements are described by positive operator-valued measures.

Dynamics of a closed system is a strongly continuous one-parameter unitary group $U(t)$. By Stone's theorem $U(t) = e^{-iHt/\hbar}$ for a unique self-adjoint $H$, and on the domain of $H$ this gives the Schrödinger equation. In the Schrödinger picture $\rho(t) = U(t)\rho U(t)^\dagger$; in the Heisenberg picture $A(t) = U(t)^\dagger A U(t)$, which obeys $\frac{dA}{dt} = \frac{i}{\hbar}[H,A]$ on suitable domains. Self-adjointness matters: a merely symmetric Hamiltonian does not define a unitary evolution unless it has a self-adjoint extension, and different extensions are different physics (for example different boundary conditions).

Symmetries are bijections of $\mathbb{P}(\mathcal{H})$ preserving transition probabilities $|\langle\phi|\psi\rangle|^2$. Wigner's theorem says each is induced by a unitary or an antiunitary operator, unique up to a phase. A symmetry group therefore acts by a projective unitary representation, which is why spin $1/2$ appears through $SU(2)$ rather than $SO(3)$.

Canonical structure. For a particle in $\mathbb{R}^n$ one needs self-adjoint $Q_j, P_k$ with $[Q_j,P_k] = i\hbar\,\delta_{jk}$. Since this relation cannot hold for bounded operators, one uses the Weyl form $e^{isQ_j}e^{itP_k} = e^{-i\hbar st\,\delta_{jk}}e^{itP_k}e^{isQ_j}$. The Stone-von Neumann theorem states that every irreducible strongly continuous representation of these relations is unitarily equivalent to the Schrödinger representation on $L^2(\mathbb{R}^n)$. This uniqueness fails for infinitely many degrees of freedom, which is one source of the extra structure of quantum field theory.

Composite systems carry the tensor product $\mathcal{H}_A\otimes\mathcal{H}_B$, and the state of a subsystem is the partial trace $\rho_A = \operatorname{tr}_B\rho$.

An equivalent algebraic formulation starts from a C*-algebra $\mathfrak{A}$ of observables and defines states as positive normalized linear functionals $\omega$ on $\mathfrak{A}$. The GNS construction recovers a Hilbert space representation from each state. Classical mechanics is the case of a commutative algebra, such as functions on phase space, and quantization replaces it by a noncommutative one.

Formulations

There are different ways of describing quantum mechanics. Each has its individual strength and weaknesses, but in terms of observable predictions they are all equivalent. This situation is analogous to how we can describe classical mechanics in terms of Newtonian mechanics, Lagrangian mechanics, Hamiltonian mechanics or Koopman-von-Neumann mechanics.

The transition from classical mechanics to quantum mechanics is known as quantization.

The most famous descriptions of quantum mechanics are

• Canonical quantum mechanics $\leftrightarrow$ quantum mechanics in Hilbert space
  • Schrödinger's wave mechanics
  • Heisenberg's matrix mechanics
• Path-integral quantum mechanics $\leftrightarrow$ quantum mechanics in configuration space
• Phase space quantum mechanics $\leftrightarrow$ quantum mechanics in phase space
• Bohmian mechanics $\leftrightarrow$ quantum mechanics in everyday space

Both Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulations both belong to the description known as canonical quantum mechanics. The relevant mathematical stage for both formulations is Hilbert space. The connection between them lies in the identification of Heisenberg's infinite matrices $p_j$ and $q^i$ ($i,j=1,2,3$), representing the momentum and position of a particle moving in $\mathbb{R}^3$, with Schrödinger's operators $-i\hbar\partial/\partial x^j$ and $x^i$ (seen as a multiplication operator) on the Hilbert space $\mathcal H=L^2(\mathbb{R}^3)$, respectively. The key to this identification lies in the canonical commutation relations $$[p_i,q^j]=-i\hbar \delta^j_i.$$ We usually call these two formulations the "Heisenberg picture" and the "Schrödinger picture", since, both descriptions are actually equivalent. In some sense, the transformation between them is "just a basis change in Hilbert space"¹.

Instead of using a Hilbert space, we can use the corresponding configuration space and its associated tangent bundle. In this formulation, our main object is the Lagrangian. This Lagrangian formulation of quantum mechanics is known as path integral quantum mechanics.

Another possibility is to use the corresponding phase space. Our main interest in phase space is the Hamiltonian and the algebraic structure of quantum mechanics is given by the Moyal bracket. The Moyal bracket is the appropriate deformation of the Poisson bracket, which governs the phase space formulation of classical mechanics, i.e. Hamiltonian mechanics. This way of describing quantum mechanics is simply known as phase space formulation of quantum mechanics.

Finally, we can also focus on the individual trajectories in real space ($\mathbb{R}^3$) of the objects in our system. This formulation of quantum mechanics is known as Bohmian mechanics and is analogous to the Newtonian formulation of classical mechanics.

• For an interesting alternative introduction to quantum mechanics, see PHYS771 Lecture 9: Quantum.
• see also https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics

Interpretations

There is no general consensus as to what the fundamental principles of quantum mechanics are and what it really "means". While almost any physicist can do calculations² in quantum mechanics, the stories that are told about what we really do when we perform these calculations vary wildly. For example, a common question is whether a particle in quantum mechanics already has well-defined properties before we measure it or if they only take on definite values as soon as we measure them.

The thing is that experimentally outcomes stay the same no matter which interpretation we believe in³. In this sense, discussions about the interpretation of quantum mechanics are mostly a matter of taste.

Important notions regarding the interpretation of quantum mechanics are

• the EPR paradox,
• Bell's theorem,
• the no-clone theorem,
• Schrödinger's cat,
• the quantum Zeno paradox.

The standard (orthodox) interpretation of quantum mechanics is presented in almost every textbook and known as the Copenhagen interpretation.

According to this interpretation, particles do not possess specific dynamical properties (momentum, position, angular momentum, energy, etc.) until we perform a measurement.

The wave function is interpreted statistically and it collapses once we measure it. Therefore, if we immediately repeat a measurement, we will get the same result again.

Regarding the question, whether a particle already has a definite momentum etc. before we measure it, the Copenhagen interpretation states that

"observations not only disturb what has to be measured, they produce it!" - Pascual Jordan.

In contrast, hidden variable interpretations which are also called realist interpretations, state that

“the position of the particle was never indeterminate, but was merely unknown to the experimenter.” - Bernard d'Espagnat.

A third popular interpretation is called the agnostic interpretation states that it makes no sense to ask such a question since how can we discuss anything that we can never measure. By definition, a property like momentum is undetermined until we measure it and a discussion about its value before measurement makes no sense:

"One should no more rack one’s brain about the problem of whether something one cannot know anything about exists all the same, than about the ancient question of how many angels are able to sit on the point of a needle." - Wolfgang Pauli

An amazing discussion of the Copenhagen interpreation and how it came about can be found in Quantum Dialogue by Mara Beller.

There are dozens of other interpretations of what quantum mechanics really means:

• Hidden-Variable Interpretations
  • Bohmian Mechanics
• Stochastic Interpretation
  • "It is well known that under certain restrictions, quantum mechanics can be interpreted as a Markov process for classical coordinates. Various authors have derived Schrodinger's equation within the context of classical diffusion theory. These works interpret quantum statistics or quantum indeterminacy as originating from the interaction of nonrelativistic point particles with random forces. These forces must be present even in the vacuum. This view does not conflict with present understanding of nature, since vacuum fluctuations in quantum field theory are known to exist and are well verified experimentally. The random forces experienced by particles in the stochastic interpretation may be viewed as coming about due to random fluctuations of the various fields in the vacuum." ⁴
  • Review of stochastic mechanics by Edward Nelson
• Dynamical Collapse Interpretations
  • https://plato.stanford.edu/entries/qm-collapse/
• Quantum Bayesianism
  • The Interpretation of Quantum Mechanics by Tom Banks
• Everett (many-worlds) Interpretation
  • "Everett's thesis on the many worlds interpretation of quantum mechanics, which is an attempt to reformulate quantum mechanics using Shannon information tools."⁵
  • Why the Many-Worlds Formulation of Quantum Mechanics Is Probably Correct by Sean Carroll
• Relational Quantum Mechanics
  • What is quantum mechanics trying to tell us? by N. David Mermin
  • Relational Quantum Mechanics by Carlo Rovelli
• Quantum Mechanics Without State Vectors
  • The Trouble with Quantum Mechanics by Steven Weinberg
  • Quantum Mechanics Without State Vectors by Steven Weinberg
• The Cellular Automaton Interpretation of Quantum Mechanics
  • The Cellular Automaton Interpretation of Quantum Mechanics by Gerard 't Hooft
• Consistent histories interpretation
  • Consistent Quantum Theory by Robert B. Griffiths
• Shut up and calculate
  • This is basically the agnostic approach where we say that since the interpretation doesn't matter we shouldn't spent time discussing about it.
  • Shut up and calculate by Max Tegmark

Recommended Resources:

• For a quick overview, see http://www.preposterousuniverse.com/blog/2014/05/29/quantum-mechanics-smackdown/
• For an overview see https://arxiv.org/pdf/1703.08341.pdf
• A good book on the topic is Roland Omnes, Interpretation of Quantum Mechanics, Princeton U. Press, Princeton, 1994.
• See also Elegance and Enigma - The Quantum Interviews by Schlosshauer
• Making Sense of Quantum Mechanics by Bricmont

“If you are not confused by quantum mechanics, then you haven’t really understood it.” Niels Bohr

“I think I can safely say that nobody understands quantum mechanics.” Richard Feynman

Roadmaps

History

In the beginning quantum mechanics was only a set of heuristic rules derived from experimental observations like the infrared catastrophe in the black-body radation. This set of heuristic rules is known as the "Old Quantum Theory"⁶.

Historically, Heisenberg's matrix mechanics⁷ was the first complete formulation of quantum mechanics. Heisenberg's main focus was the noncommutative structure of the quantum mechanical algebra of observables.

Soon after, Schrödinger developed his wave mechanics⁸ whose main focus is the classical geometric structure of configuration space and all about wave functions.

Only one year afterward, Neumann⁹ unified the two approaches by introducing the abstract concept of a Hilbert space. Schrödinger's wave functions are vectors living in Hilbert space and Heisenberg’s observables are linear operators acting on these vectors.

Heisenberg's matrix mechanics was developed into its current complete full set of equations by Born, Jordan, and Heisenberg ¹⁰

Around the same time, Dirac discovered independently the same structure¹¹).

The spectrum of the Hydrogen atom was first calculated by Pauli¹².

Only some time afterward - in 1926 - Erwin Schrödinger introduced the concept of the wave function and also calculated the hydrogen spectrum. Since in principle, quantum mechanics was already complete before Schrödinger's famous paper it is reasonable to ask: "So, what did Schrödinger do, in his 1926 paper?"

With hindsight, he took a technical and a conceptual step. The technical step was to change the algebraic language of the theory, unfamiliar at the time, into a familiar one: differential equations. This brought ethereal quantum theory down to the level of the average theoretical physicist. The conceptual step was to introduce the notion of “wave function” ψ, soon to be evolved into the notion of “quantum state” ψ, endowing it with heavy ontological weight."Space is blue and birds fly through it" by Carlo Rovelli

Since Heisenberg and Co. were much earlier than Schrödinger it also makes sense to ask why ultimately Schrödinger "won". Conventionally Students are introduced to quantum mechanics by starting with Schrödinger's "wave mechanics".

Heisenberg lost the political battle against Schrödinger, for a number of reasons. First, all this was about “interpretation” and for many physicists this wasn’t so interesting after all, once the equations of quantum mechanics begun producing wonders. Differential equations are easier to work with and sort of visualise, than non-commutative algebras. Third, Dirac himself, who did a lot directly with non-commutative algebras, found it easier to make the calculus concrete by giving it a linear representation on Hilbert spaces, and von Neumann followed: on the one hand, his robust mathematical formulation of the theory brilliantly focused on the proper relevant notion: the non-commutative observable algebra, on the other, the weight given to the Hilbert space could be taken by some as an indirect confirmation of the ontological weight of the quantum states. Fourth, and most importantly, Bohr —the recognised fatherly figure of the community— tried to mediate between his two brilliant bickering children, by obscurely agitating hands about a shamanic “wave/particle duality”. "Space is blue and birds fly through it" by Carlo Rovelli

Recommended Resources

• Jim Baggott; The Quantum Story
• Robert P. Crease and Charles C. Mann, The Second Creation: Makers of the Revolution in Twentieth-Century Physics
• Abraham Pais, Inward Bound: of Matter and Forces in the Physical World
• Uncertainty by David Lindley
• See also Ron Maimon's answer, Good book on the history of Quantum Mechanics?, URL (version: 2011-12-23):
• W. a. Fedak and J. J. Prentis, “The 1925 Born and Jordan paper ’On quantum mechanics’,” American Journal of Physics 77 (2009) 128.
• B. van der Waerden, Sources of quantum mechanics. North Holland, 1967.

FAQ

For many more questions and answers see: http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Hilbert Space
6. Legendre Transformation
7. Newton's Second Law
8. Vector Calculus
9. Newtonian Mechanics
10. Variational Calculus
11. Configuration Space
12. Euler-Lagrange Equations
13. Lagrangian Formalism
14. Phase Space
15. Hamiltonian Formalism
16. Schrödinger Equation
17. Canonical Quantum Mechanics