Examples

For an equation describing a physical phenomenon, the superposition principle states that a combination of solutions to a linear equation is also a solution of it. When this is true the equation is said to obey the superposition principle. Thus, if state vectors f₁, f₂ and f₃ each solve the linear equation on ψ, then ψ = c₁ f₁ + c₂ f₂ + c₃ f₃ would also be a solution, in which each c is a coefficient. The Schrödinger equation is linear, so quantum mechanics follows this.

For example, consider an electron with two possible configurations: up and down. This describes the physical system of a qubit.

${\displaystyle c_{1}{\mid }{\uparrow }\rangle +c_{2}{\mid }{\downarrow }\rangle }$

is the most general state. But these coefficients dictate probabilities for the system to be in either configuration. The probability for a specified configuration is given by the square of the absolute value of the coefficient. The probabilities must add to 1, since the electron must be in one of those two states.

${\displaystyle p_{\text{up}}={\mid }c_{1}{\mid }^{2}}$

${\displaystyle p_{\text{down}}={\mid }c_{2}\mid ^{2}}$

${\displaystyle p_{\text{up or down}}=p_{\text{up}}+p_{\text{down}}=1}$

Continuing with this example, if a particle can be in state  up and  down, it can also be in a state where it is an amount 3i/5 up and an amount 4/5 down.

${\displaystyle |\psi \rangle ={3 \over 5}i{\mid }{\uparrow }\rangle +{4 \over 5}{\mid }{\downarrow }\rangle .}$

In this, the probability for up is ${\displaystyle \left|{\frac {3i}{5}}\right|^{2}={\frac {9}{25}}}$. The probability for down is ${\displaystyle \left|{\frac {4}{5}}\right|^{2}={\frac {16}{25}}}$. Note that ${\displaystyle {\frac {9}{25}}+{\frac {16}{25}}=1}$.

In the description, only the relative size of the different components matter, and their angle to each other on the complex plane. This is usually stated by declaring that two states which are a multiple of one another are the same as far as the description of the situation is concerned. Either of these describe the same state for any nonzero ${\displaystyle \alpha }$

${\displaystyle |\psi \rangle \approx \alpha |\psi \rangle }$

The fundamental law of quantum mechanics is that the evolution is linear, meaning that if state A turns into A′ and B turns into B′ after 10 seconds, then after 10 seconds the superposition ${\displaystyle \psi }$ turns into a mixture of A′ and B′ with the same coefficients as A and B.

For example, if we have the following

${\displaystyle {\mid }{\uparrow }\rangle \to {\mid }{\downarrow }\rangle }$

${\displaystyle {\mid }{\downarrow }\rangle \to {\frac {3i}{5}}{\mid }{\uparrow }\rangle +{\frac {4}{5}}{\mid }{\downarrow }\rangle }$

Then after those 10 seconds our state will change to

${\displaystyle c_{1}{\mid }{\uparrow }\rangle +c_{2}{\mid }{\downarrow }\rangle \to c_{1}\left({\mid }{\downarrow }\rangle \right)+c_{2}\left({\frac {3i}{5}}{\mid }{\uparrow }\rangle +{\frac {4}{5}}{\mid }{\downarrow }\rangle \right)}$

So far there have just been 2 configurations, but there can be infinitely many.

In illustration, a particle can have any position, so that there are different configurations which have any value of the position x. These are written:

${\displaystyle |x\rangle }$

The principle of superposition guarantees that there are states which are arbitrary superpositions of all the positions with complex coefficients:

${\displaystyle \sum _{x}\psi (x)|x\rangle }$

This sum is defined only if the index x is discrete. If the index is over ${\displaystyle \mathbb {R} }$, then the sum is replaced by an integral. The quantity ${\displaystyle \psi (x)}$ is called the wavefunction of the particle.

If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:

${\displaystyle \sum _{x}\psi _{+}(x)|x,{\uparrow }\rangle +\psi _{-}(x)|x,{\downarrow }\rangle \,}$

The configuration space of a quantum mechanical system cannot be worked out without some physical knowledge. The input is usually the allowed different classical configurations, but without the duplication of including both position and momentum.

A pair of particles can be in any combination of pairs of positions. A state where one particle is at position x and the other is at position y is written ${\displaystyle |x,y\rangle }$. The most general state is a superposition of the possibilities:

${\displaystyle \sum _{xy}A(x,y)|x,y\rangle \,}$

The description of the two particles is much larger than the description of one particle—it is a function in twice the number of dimensions. This is also true in probability, when the statistics of two random variables are correlated. If two particles are uncorrelated, the probability distribution for their joint position P(x, y) is a product of the probability of finding one at one position and the other at the other position:

${\displaystyle P(x,y)=P_{x}(x)P_{y}(y)\,}$

This means that the wave function ${\displaystyle A(x,y)}$ of the system can be represented as a product of the wave functions ${\displaystyle \psi _{x}(x)}$ and ${\displaystyle \psi _{y}(y)}$ of its parts:^[3]

${\displaystyle A(x,y)=\psi _{x}(x)\psi _{y}(y)\,}$.

In 1927, Heitler and London,^[4] attempted to quantitatively mechanically calculate the ground steady state of the H₂ molecule. The calculations were based on the quantum superposition of the two hydrogen atoms that make up the system - H₂ molecule. The success of this attempt became the basis for all further development of covalent bond.

Analogy with probability

In probability theory there is a similar principle. If a system has a probabilistic description, this description gives the probability of any configuration, and given any two different configurations, there is a state which is partly this and partly that, with positive real number coefficients, the probabilities, which say how much of each there is.

For example, if we have a probability distribution for where a particle is, it is described by the "state"

${\displaystyle \sum _{x}\rho (x)|x\rangle }$

Where ${\displaystyle \rho }$ is the probability density function, a positive number that measures the probability that the particle will be found at a certain location.

The evolution equation is also linear in probability, for fundamental reasons. If the particle has some probability for going from position x to y, and from z to y, the probability of going to y starting from a state which is half-x and half-z is a half-and-half mixture of the probability of going to y from each of the options. This is the principle of linear superposition in probability.

Quantum mechanics is different, because the numbers can be positive or negative. While the complex nature of the numbers is just a doubling, if you consider the real and imaginary parts separately, the sign of the coefficients is important. In probability, two different possible outcomes always add together, so that if there are more options to get to a point z, the probability always goes up. In quantum mechanics, different possibilities can cancel.

In probability theory with a finite number of states, the probabilities can always be multiplied by a positive number to make their sum equal to one. For example, if there is a three state probability system:

${\displaystyle x|1\rangle +y|2\rangle +z|3\rangle \,}$

where the probabilities ${\displaystyle x,y,z}$ are positive numbers. Rescaling x,y,z so that

${\displaystyle x+y+z=1\,}$

The geometry of the state space is a revealed to be a triangle. In general it is a simplex. There are special points in a triangle or simplex corresponding to the corners, and these points are those where one of the probabilities is equal to 1 and the others are zero. These are the unique locations where the position is known with certainty.

In a quantum mechanical system with three states, the quantum mechanical wavefunction is a superposition of states again, but this time twice as many quantities with no restriction on the sign:

${\displaystyle A|1\rangle +B|2\rangle +C|3\rangle =(A_{r}+iA_{i})|1\rangle +(B_{r}+iB_{i})|2\rangle +(C_{r}+iC_{i})|3\rangle \,}$

rescaling the variables so that the sum of the squares is 1, the geometry of the space is revealed to be a high-dimensional sphere

${\displaystyle A_{r}^{2}+A_{i}^{2}+B_{r}^{2}+B_{i}^{2}+C_{r}^{2}+C_{i}^{2}=1\,}$.

A sphere has a large amount of symmetry, it can be viewed in different coordinate systems or bases. So unlike a probability theory, a quantum theory has a large number of different bases in which it can be equally well described. The geometry of the phase space can be viewed as a hint that the quantity in quantum mechanics which corresponds to the probability is the absolute square of the coefficient of the superposition.

Hamiltonian evolution

The numbers that describe the amplitudes for different possibilities define the kinematics, the space of different states. The dynamics describes how these numbers change with time. For a particle that can be in any one of infinitely many discrete positions, a particle on a lattice, the superposition principle tells you how to make a state:

${\displaystyle \sum _{n}\psi _{n}|n\rangle \,}$

So that the infinite list of amplitudes ${\textstyle (\ldots ,\psi _{-2},\psi _{-1},\psi _{0},\psi _{1},\psi _{2},\ldots )}$ completely describes the quantum state of the particle. This list is called the state vector, and formally it is an element of a Hilbert space, an infinite-dimensional complex vector space. It is usual to represent the state so that the sum of the absolute squares of the amplitudes is one:

${\displaystyle \sum \psi _{n}^{*}\psi _{n}=1}$

For a particle described by probability theory random walking on a line, the analogous thing is the list of probabilities ${\textstyle (\ldots ,P_{-2},P_{-1},P_{0},P_{1},P_{2},\ldots )}$, which give the probability of any position. The quantities that describe how they change in time are the transition probabilities ${\displaystyle \scriptstyle K_{x\rightarrow y}(t)}$, which gives the probability that, starting at x, the particle ends up at y time t later. The total probability of ending up at y is given by the sum over all the possibilities

${\displaystyle P_{y}(t_{0}+t)=\sum _{x}P_{x}(t_{0})K_{x\rightarrow y}(t)\,}$

The condition of conservation of probability states that starting at any x, the total probability to end up somewhere must add up to 1:

${\displaystyle \sum _{y}K_{x\rightarrow y}=1\,}$

So that the total probability will be preserved, K is what is called a stochastic matrix.

When no time passes, nothing changes: for 0 elapsed time ${\displaystyle \scriptstyle K{x\rightarrow y}(0)=\delta _{xy}}$, the K matrix is zero except from a state to itself. So in the case that the time is short, it is better to talk about the rate of change of the probability instead of the absolute change in the probability.

${\displaystyle P_{y}(t+dt)=P_{y}(t)+dt\,\sum _{x}P_{x}R_{x\rightarrow y}\,}$

where ${\displaystyle \scriptstyle R_{x\rightarrow y}}$ is the time derivative of the K matrix:

${\displaystyle R_{x\rightarrow y}={K_{x\rightarrow y}\,dt-\delta _{xy} \over dt}.\,}$

The equation for the probabilities is a differential equation that is sometimes called the master equation:

${\displaystyle {dP_{y} \over dt}=\sum _{x}P_{x}R_{x\rightarrow y}\,}$

The R matrix is the probability per unit time for the particle to make a transition from x to y. The condition that the K matrix elements add up to one becomes the condition that the R matrix elements add up to zero:

${\displaystyle \sum _{y}R_{x\rightarrow y}=0\,}$

One simple case to study is when the R matrix has an equal probability to go one unit to the left or to the right, describing a particle that has a constant rate of random walking. In this case ${\displaystyle \scriptstyle R_{x\rightarrow y}}$ is zero unless y is either x + 1, x, or x − 1, when y is x + 1 or x − 1, the R matrix has value c, and in order for the sum of the R matrix coefficients to equal zero, the value of ${\displaystyle R_{x\rightarrow x}}$ must be −2c. So the probabilities obey the discretized diffusion equation:

${\displaystyle {dP_{x} \over dt}=c(P_{x+1}-2P_{x}+P_{x-1})\,}$

which, when c is scaled appropriately and the P distribution is smooth enough to think of the system in a continuum limit becomes:

${\displaystyle {\partial P(x,t) \over \partial t}=c{\partial ^{2}P \over \partial x^{2}}\,}$

Which is the diffusion equation.

Quantum amplitudes give the rate at which amplitudes change in time, and they are mathematically exactly the same except that they are complex numbers. The analog of the finite time K matrix is called the U matrix:

${\displaystyle \psi _{n}(t)=\sum _{m}U_{nm}(t)\psi _{m}\,}$

Since the sum of the absolute squares of the amplitudes must be constant, ${\displaystyle U}$ must be unitary:

${\displaystyle \sum _{n}U_{nm}^{*}U_{np}=\delta _{mp}\,}$

or, in matrix notation,

${\displaystyle U^{\dagger }U=I\,}$

The rate of change of U is called the Hamiltonian H, up to a traditional factor of i:

${\displaystyle H_{mn}=i{d \over dt}U_{mn}}$

The Hamiltonian gives the rate at which the particle has an amplitude to go from m to n. The reason it is multiplied by i is that the condition that U is unitary translates to the condition:

${\displaystyle (I+iH^{\dagger }\,dt)(I-iH\,dt)=I}$

${\displaystyle H^{\dagger }-H=0\,}$

which says that H is Hermitian. The eigenvalues of the Hermitian matrix H are real quantities, which have a physical interpretation as energy levels. If the factor i were absent, the H matrix would be antihermitian and would have purely imaginary eigenvalues, which is not the traditional way quantum mechanics represents observable quantities like the energy.

For a particle that has equal amplitude to move left and right, the Hermitian matrix H is zero except for nearest neighbors, where it has the value c. If the coefficient is everywhere constant, the condition that H is Hermitian demands that the amplitude to move to the left is the complex conjugate of the amplitude to move to the right. The equation of motion for ${\displaystyle \psi }$ is the time differential equation:

${\displaystyle i{d\psi _{n} \over dt}=c^{*}\psi _{n+1}+c\psi _{n-1}}$

In the case in which left and right are symmetric, c is real. By redefining the phase of the wavefunction in time, ${\displaystyle \psi \rightarrow \psi e^{i2ct}}$, the amplitudes for being at different locations are only rescaled, so that the physical situation is unchanged. But this phase rotation introduces a linear term.

${\displaystyle i{d\psi _{n} \over dt}=c\psi _{n+1}-2c\psi _{n}+c\psi _{n-1},}$

which is the right choice of phase to take the continuum limit. When ${\displaystyle c}$ is very large and ${\displaystyle \psi }$ is slowly varying so that the lattice can be thought of as a line, this becomes the free Schrödinger equation:

${\displaystyle i{\partial \psi \over \partial t}=-{\partial ^{2}\psi \over \partial x^{2}}}$

If there is an additional term in the H matrix that is an extra phase rotation that varies from point to point, the continuum limit is the Schrödinger equation with a potential energy:

${\displaystyle i{\partial \psi \over \partial t}=-{\partial ^{2}\psi \over \partial x^{2}}+V(x)\psi }$

These equations describe the motion of a single particle in non-relativistic quantum mechanics.

Quantum mechanics in imaginary time

The analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. In a statistical system in discrete time, t=1,2,3, described by a transition matrix for one time step ${\displaystyle \scriptstyle K_{m\rightarrow n}}$, the probability to go between two points after a finite number of time steps can be represented as a sum over all paths of the probability of taking each path:

${\displaystyle K_{x\rightarrow y}(T)=\sum _{x(t)}\prod _{t}K_{x(t)x(t+1)}\,}$

where the sum extends over all paths ${\displaystyle x(t)}$ with the property that ${\displaystyle x(0)=0}$ and ${\displaystyle x(T)=y}$. The analogous expression in quantum mechanics is the path integral.

A generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a nonzero probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions. In probability theory, the probability m for the stochastic matrix obeys detailed balance when the stationary distribution ${\displaystyle \rho _{n}}$ has the property:

${\displaystyle \rho _{n}K_{n\rightarrow m}=\rho _{m}K_{m\rightarrow n}\,}$

Detailed balance says that the total probability of going from m to n in the stationary distribution, which is the probability of starting at m ${\displaystyle \rho _{m}}$ times the probability of hopping from m to n, is equal to the probability of going from n to m, so that the total back-and-forth flow of probability in equilibrium is zero along any hop. The condition is automatically satisfied when n=m, so it has the same form when written as a condition for the transition-probability R matrix.

${\displaystyle \rho _{n}R_{n\rightarrow m}=\rho _{m}R_{m\rightarrow n}\,}$

When the R matrix obeys detailed balance, the scale of the probabilities can be redefined using the stationary distribution so that they no longer sum to 1:

${\displaystyle p'_{n}={\sqrt {\rho _{n}}}\;p_{n}\,}$

In the new coordinates, the R matrix is rescaled as follows:

${\displaystyle {\sqrt {\rho _{n}}}R_{n\rightarrow m}{1 \over {\sqrt {\rho _{m}}}}=H_{nm}\,}$

and H is symmetric

${\displaystyle H_{nm}=H_{mn}\,}$

This matrix H defines a quantum mechanical system:

${\displaystyle i{d \over dt}\psi _{n}=\sum H_{nm}\psi _{m}\,}$

whose Hamiltonian has the same eigenvalues as those of the R matrix of the statistical system. The eigenvectors are the same too, except expressed in the rescaled basis. The stationary distribution of the statistical system is the ground state of the Hamiltonian and it has energy exactly zero, while all the other energies are positive. If H is exponentiated to find the U matrix:

${\displaystyle U(t)=e^{-iHt}\,}$

and t is allowed to take on complex values, the K' matrix is found by taking time imaginary.

${\displaystyle K'(t)=e^{-Ht}\,}$

For quantum systems which are invariant under time reversal the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation. If such a Hamiltonian has a unique lowest energy state with a positive real wave-function, as it often does for physical reasons, it is connected to a stochastic system in imaginary time. This relationship between stochastic systems and quantum systems sheds much light on supersymmetry.

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