Quantum mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.

Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale.

Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values (quantization), objects have characteristics of both particles and waves (wave-particle duality), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle).

Mathematical formulation

In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ${\displaystyle \psi }$ belonging to a (separable) complex Hilbert space ${\displaystyle {\mathcal {H}}}$. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys ${\displaystyle \langle \psi ,\psi \rangle =1}$, and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ${\displaystyle \psi }$ and ${\displaystyle e^{i\alpha }\psi }$ represent the same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions ${\displaystyle L^{2}(\mathbb {C} )}$, while the Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors ${\displaystyle \mathbb {C} ^{2}}$ with the usual inner product.

Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint) linear operators acting on the Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition. When an observable is measured, the result will be one of its eigenvalues with probability given by the Born rule: in the simplest case the eigenvalue ${\displaystyle \lambda }$ is non-degenerate and the probability is given by ${\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}}$, where ${\displaystyle {\vec {\lambda }}}$ is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ${\displaystyle \langle \psi ,P_{\lambda }\psi \rangle }$, where ${\displaystyle P_{\lambda }}$ is the projector onto its associated eigenspace. In the continuous case, these formulas give instead the probability density.

After the measurement, if result ${\displaystyle \lambda }$ was obtained, the quantum state is postulated to collapse to ${\displaystyle {\vec {\lambda }}}$, in the non-degenerate case, or to ${\displaystyle P_{\lambda }\psi /{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}}$, in the general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of "wave function collapse" (see, for example, the many-worlds interpretation). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics.

The time evolution of a quantum state is described by the Schrödinger equation:

${\displaystyle i\hbar {\frac {d}{dt}}\psi (t)=H\psi (t).}$

Here ${\displaystyle H}$ denotes the Hamiltonian, the observable corresponding to the total energy of the system, and ${\displaystyle \hbar }$ is the reduced Planck constant. The constant ${\displaystyle i\hbar }$ is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle.

The solution of this differential equation is given by

${\displaystyle \psi (t)=e^{-iHt/\hbar }\psi (0).}$

The operator ${\displaystyle U(t)=e^{-iHt/\hbar }}$ is known as the time-evolution operator, and has the crucial property that it is unitary. This time evolution is deterministic in the sense that – given an initial quantum state ${\displaystyle \psi (0)}$  – it makes a definite prediction of what the quantum state ${\displaystyle \psi (t)}$ will be at any later time.

Fig. 1: Probability densities corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: n = 1, 2, 3, ...) and angular momenta (increasing across from left to right: s, p, d, ...). Denser areas correspond to higher probability density in a position measurement. Such wave functions are directly comparable to Chladni's figures of acoustic modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp energy and thus, a definite frequency. The angular momentum and energy are quantized and take only discrete values like those shown (as is the case for resonant frequencies in acoustics)

Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital 

Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator, the particle in a box, the dihydrogen cation, and the hydrogen atom. Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment.

However, there are techniques for finding approximate solutions. One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy. Another method is called "semi-classical equation of motion", which applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. This approach is particularly important in the field of quantum chaos.

Uncertainty principle

One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observables, meaning that they are represented by Hermitian operators. The position operator ${\displaystyle {\hat {X}}}$ and momentum operator ${\displaystyle {\hat {P}}}$ do not commute, but rather satisfy the canonical commutation relation:

${\displaystyle [{\hat {X}},{\hat {P}}]=i\hbar .}$

Given a quantum state, the Born rule lets us compute expectation values for both ${\displaystyle X}$ and ${\displaystyle P}$, and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation, we have

${\displaystyle \sigma _{X}={\sqrt {\langle {X}^{2}\rangle -\langle {X}\rangle ^{2}}},}$

and likewise for the momentum:

${\displaystyle \sigma _{P}={\sqrt {\langle {P}^{2}\rangle -\langle {P}\rangle ^{2}}}.}$

The uncertainty principle states that

${\displaystyle \sigma _{X}\sigma _{P}\geq {\frac {\hbar }{2}}.}$

Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators ${\displaystyle A}$ and ${\displaystyle B}$. The commutator of these two operators is

${\displaystyle [A,B]=AB-BA,}$

and this provides the lower bound on the product of standard deviations:

${\displaystyle \sigma _{A}\sigma _{B}\geq {\frac {1}{2}}\left|\langle [A,B]\rangle \right|.}$

Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an ${\displaystyle i/\hbar }$ factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space. This is why in quantum equations in position space, the momentum ${\displaystyle p_{i}}$ is replaced by ${\displaystyle -i\hbar {\frac {\partial }{\partial x}}}$, and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times ${\displaystyle -\hbar ^{2}}$.

Composite systems and entanglement

When two different quantum systems are considered together, the Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces ${\displaystyle {\mathcal {H}}_{A}}$ and ${\displaystyle {\mathcal {H}}_{B}}$, respectively. The Hilbert space of the composite system is then

${\displaystyle {\mathcal {H}}_{AB}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}.}$

If the state for the first system is the vector ${\displaystyle \psi _{A}}$ and the state for the second system is ${\displaystyle \psi _{B}}$, then the state of the composite system is

${\displaystyle \psi _{A}\otimes \psi _{B}.}$

Not all states in the joint Hilbert space ${\displaystyle {\mathcal {H}}_{AB}}$ can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ${\displaystyle \psi _{A}}$ and ${\displaystyle \phi _{A}}$ are both possible states for system ${\displaystyle A}$, and likewise ${\displaystyle \psi _{B}}$ and ${\displaystyle \phi _{B}}$ are both possible states for system ${\displaystyle B}$, then

${\displaystyle {\tfrac {1}{\sqrt {2}}}\left(\psi _{A}\otimes \psi _{B}+\phi _{A}\otimes \phi _{B}\right)}$

is a valid joint state that is not separable. States that are not separable are called entangled.

If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.

As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence. This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.

Equivalence between formulations

There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "transformation theory" proposed by Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger). An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics.

Symmetries and conservation laws

The Hamiltonian ${\displaystyle H}$ is known as the generator of time evolution, since it defines a unitary time-evolution operator ${\displaystyle U(t)=e^{-iHt/\hbar }}$ for each value of ${\displaystyle t}$. From this relation between ${\displaystyle U(t)}$ and ${\displaystyle H}$, it follows that any observable ${\displaystyle A}$ that commutes with ${\displaystyle H}$ will be conserved: its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator ${\displaystyle A}$ can generate a family of unitary operators parameterized by a variable ${\displaystyle t}$. Under the evolution generated by ${\displaystyle A}$, any observable ${\displaystyle B}$ that commutes with ${\displaystyle A}$ will be conserved. Moreover, if ${\displaystyle B}$ is conserved by evolution under ${\displaystyle A}$, then ${\displaystyle A}$ is conserved under the evolution generated by ${\displaystyle B}$. This implies a quantum version of the result proven by Emmy Noether in classical (Lagrangian) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law.

Examples

Free particle

Position space probability density of a Gaussian wave packet moving in one dimension in free space.

The simplest example of quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy:

${\displaystyle H={\frac {1}{2m}}P^{2}=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}.}$

The general solution of the Schrödinger equation is given by

${\displaystyle \psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {\psi }}(k,0)e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}\mathrm {d} k,}$

which is a superposition of all possible plane waves ${\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}}$, which are eigenstates of the momentum operator with momentum ${\displaystyle p=\hbar k}$. The coefficients of the superposition are ${\displaystyle {\hat {\psi }}(k,0)}$, which is the Fourier transform of the initial quantum state ${\displaystyle \psi (x,0)}$.

It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian wave packet:

${\displaystyle \psi (x,0)={\frac {1}{\sqrt[{4}]{\pi a}}}e^{-{\frac {x^{2}}{2a}}}}$

which has Fourier transform, and therefore momentum distribution

${\displaystyle {\hat {\psi }}(k,0)={\sqrt[{4}]{\frac {a}{\pi }}}e^{-{\frac {ak^{2}}{2}}}.}$

We see that as we make ${\displaystyle a}$ smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making ${\displaystyle a}$ larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle.

As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.

Particle in a box

1-dimensional potential energy box (or infinite potential well)

The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region. For the one-dimensional case in the ${\displaystyle x}$ direction, the time-independent Schrödinger equation may be written

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .}$

With the differential operator defined by

${\displaystyle {\hat {p}}_{x}=-i\hbar {\frac {d}{dx}}}$

the previous equation is evocative of the classic kinetic energy analogue,

${\displaystyle {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,}$

with state ${\displaystyle \psi }$ in this case having energy ${\displaystyle E}$ coincident with the kinetic energy of the particle.

The general solutions of the Schrödinger equation for the particle in a box are

${\displaystyle \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}}$

or, from Euler's formula,

${\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).\!}$

The infinite potential walls of the box determine the values of ${\displaystyle C,D,}$ and ${\displaystyle k}$ at ${\displaystyle x=0}$ and ${\displaystyle x=L}$ where ${\displaystyle \psi }$ must be zero. Thus, at ${\displaystyle x=0}$,

${\displaystyle \psi (0)=0=C\sin(0)+D\cos(0)=D}$

and ${\displaystyle D=0}$. At ${\displaystyle x=L}$,

${\displaystyle \psi (L)=0=C\sin(kL),}$

in which ${\displaystyle C}$ cannot be zero as this would conflict with the postulate that ${\displaystyle \psi }$ has norm 1. Therefore, since ${\displaystyle \sin(kL)=0}$, ${\displaystyle kL}$ must be an integer multiple of ${\displaystyle \pi }$,

${\displaystyle k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .}$

This constraint on ${\displaystyle k}$ implies a constraint on the energy levels, yielding

${\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.}$

A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the rectangular potential barrier, which furnishes a model for the quantum tunneling effect that plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy.

Harmonic oscillator

Some trajectories of a harmonic oscillator (i.e. a ball attached to a spring) in classical mechanics (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a wave (called the wave function), with the real part shown in blue and the imaginary part shown in red. Some of the trajectories (such as C, D, E, and F) are standing waves (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy.

As in the classical case, the potential for the quantum harmonic oscillator is given by

${\displaystyle V(x)={\frac {1}{2}}m\omega ^{2}x^{2}.}$

This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The eigenstates are given by

${\displaystyle \psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad }$

${\displaystyle n=0,1,2,\ldots .}$

where H_n are the Hermite polynomials

${\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x^{2}}\right),}$

and the corresponding energy levels are

${\displaystyle E_{n}=\hbar \omega \left(n+{1 \over 2}\right).}$

This is another example illustrating the discretization of energy for bound states.

Mach–Zehnder interferometer

Schematic of a Mach–Zehnder interferometer.

The Mach–Zehnder interferometer (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the delayed choice quantum eraser, the Elitzur–Vaidman bomb tester, and in studies of quantum entanglement.

We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector ${\displaystyle \psi \in \mathbb {C} ^{2}}$ that is a superposition of the "lower" path ${\displaystyle \psi _{l}={\begin{pmatrix}1\\0\end{pmatrix}}}$ and the "upper" path ${\displaystyle \psi _{u}={\begin{pmatrix}0\\1\end{pmatrix}}}$, that is, ${\displaystyle \psi =\alpha \psi _{l}+\beta \psi _{u}}$ for complex ${\displaystyle \alpha ,\beta }$. In order to respect the postulate that ${\displaystyle \langle \psi ,\psi \rangle =1}$ we require that ${\displaystyle |\alpha |^{2}+|\beta |^{2}=1}$.

Both beam splitters are modelled as the unitary matrix ${\displaystyle B={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&i\\i&1\end{pmatrix}}}$, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of ${\displaystyle 1/{\sqrt {2}}}$, or be reflected to the other path with a probability amplitude of ${\displaystyle i/{\sqrt {2}}}$. The phase shifter on the upper arm is modelled as the unitary matrix ${\displaystyle P={\begin{pmatrix}1&0\\0&e^{i\Delta \Phi }\end{pmatrix}}}$, which means that if the photon is on the "upper" path it will gain a relative phase of ${\displaystyle \Delta \Phi }$, and it will stay unchanged if it is in the lower path.

A photon that enters the interferometer from the left will then be acted upon with a beam splitter ${\displaystyle B}$, a phase shifter ${\displaystyle P}$, and another beam splitter ${\displaystyle B}$, and so end up in the state

${\displaystyle BPB\psi _{l}=ie^{i\Delta \Phi /2}{\begin{pmatrix}-\sin(\Delta \Phi /2)\\\cos(\Delta \Phi /2)\end{pmatrix}},}$

and the probabilities that it will be detected at the right or at the top are given respectively by

${\displaystyle p(u)=|\langle \psi _{u},BPB\psi _{l}\rangle |^{2}=\cos ^{2}{\frac {\Delta \Phi }{2}},}$

${\displaystyle p(l)=|\langle \psi _{l},BPB\psi _{l}\rangle |^{2}=\sin ^{2}{\frac {\Delta \Phi }{2}}.}$

One can therefore use the Mach–Zehnder interferometer to estimate the phase shift by estimating these probabilities.

It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases there will be no interference between the paths anymore, and the probabilities are given by ${\displaystyle p(u)=p(l)=1/2}$, independently of the phase ${\displaystyle \Delta \Phi }$. From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.