- Quantum superposition
Quantum superposition is the fundamental law of
quantum mechanics. It defines the allowed state space of a quantum mechanical system.
Probability theory, every possible event has a positive numberassociated to it, the probability, which gives the chance that it happens. If you want to know the probability of two independent events both occurring, for example the probability that it will rain and also that the stock marketwill go up, you multiply the probability for each event. If there is a 0.3 chance that it will rain and a 0.6 chance that the market will go up, there is a 0.18 chance that both will happen.
If the market can go up because of two exclusive events, and you want to know the total probability that the market will go up, you add the probability for the two events. For example, if the market will go up if the fed lowers interest rates by 2 points or by 3 points, the probability that it will go up is the sum of the probability that the Fed lowers the interest rate by 2 points or by 3 points.
Quantum mechanics has the exact same rules for multiplying and adding numbers associated with events, except that the quantities are complex numbers called amplitudes instead of positive real numbers called probabilities. The superposition principle says that the way to describe the world is to assign such a complex number to every possible situation, and that the way to describe how things change is to treat these numbers mathematically as if they were probabilities. Because these numbers can be positive or negative, quantum mechanics allows the counterintuitive
phenomenonthat sometimes when there are more ways for a thing to happen, the chance that it happens goes down. An event with a negative amplitude can cancel with an event with a positive amplitude.
For example, if a photon in a plus spin state has a .1 amplitude to be absorbed and take an atom to the second energy level, and if the photon in a minus spin state has a -.1 amplitude to do the same thing, a photon which has an equal amplitude to be plus or minus would have zero amplitude to take the atom to the second excited state and the atom will not be excited. If the photon's spin is measured before it reaches the atom, whatever the answer, plus or minus, it will have a .1 amplitude to excite the atom.
The probability in quantum mechanics is equal to the square of the absolute value of the amplitude. The further the amplitude is from zero, the bigger the probability. In the example above, the probability that the atom will be excited is .01. But the only time probability enters the picture is when an observer gets involved. If you look to see which way the atom is, the different amplitudes become probabilities for seeing different things. So if you check to see whether the atom is excited immediately after the photon reaches it, you have a .01 chance of seeing the atom excited.
Observations have different outcomes described by probabilities, while microscopic events are described by amplitudes. This difference leads many people to wonder what the correct interpretation of the amplitude is.
The principle of superposition states that if the world can be in any configuration, any possible arrangement of particles or fields, and if the world could also be in another configuration, then the world can also be in a state which is a mixture of the two, where the amount of each configuration that is in the mixture is specified by a complex number.
For example, if a particle can be in position A and position B, it can also be in a state where it is an amount "3i/5" in position A and an amount "4/5" in position B. To write this, people usually say:
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.:
The fundamental dynamical law of quantum mechanics is that the evolution is
linear, meaning that if the state A turns into A' and B turns into B' after 10 seconds, then after 10 seconds the superposition turns into a mixture of A' and B' with the same coefficients.
A particle can have any position, so that there are different states which have any value of the position x. These are written::
The principle of superposition guarantees that there are states which are arbitrary mixtures of all the positions with complex coefficients:
The sum is usually an integral, because space is continuous. The quantity is called the
wavefunctionof the particle.
If a particle can have some discrete orientations of the spin, say the spin can be aligned with the z axis or against it , then the particle can have any state of the form::
If the particle has both position and spin, the state is a superposition of all possibilities for both:
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 . The most general state is a superposition of the possibilities:
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 things 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::
In quantum mechanics, two particles can be in special states where the amplitudes of their position are uncorrelated. For quantum amplitudes, the word entanglement replaces the word correlation, but the analogy is exact. A disentangled wavefunction has the form::
while an entangled wavefunction does not have this form. Like correlation in probability, there are many more entangled states than disentangled ones. For instance, when two particles which start out with an equal amplitude to be anywhere in a box have a strong attraction and a way to dissipate energy, they can easily come together to make a bound state. The bound state still has an equal probability to be anywhere, so that each particle is equally likely to be everywhere, but the two particles will become entangled so that wherever one particle is, the other is too.
Analogy With Probability
probability theorythere 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":Where is the
probability densityfunction, 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::where the probabilities are positive numbers. The geometry of the allowed x,y,z is a triangle shaped cone in three-dimensional space, and rescaling x,y,z so that: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::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:.
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.
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 position, a particle on a lattice, the superposition principle tells you how to make a state:
So that the infinite list of amplitudes 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 "squares" of the amplitudes add up to one::
For a particle described by probability theory random walking on a line, the analogous thing is the list of probabilities , which give the probability of any position. The quantities that describe how they change in time are the transition probabilities , which gives the probability that, starting at x, the particle ends up at y after time t. The total probability of ending up at y is given by the sum over all the possibilities
The condition of conservation of probability states that starting at any x, the total probability to end up "somewhere" must add up to 1:
So that the total probability will be preserved, K is what is called a
When no time passes, nothing changes: for zero elapsed time , 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.
where is the time derivative of the K matrix:
The equation for the probabilities is a differential equation which is sometimes called the master equation:
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:
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 which has a constant rate of random walking. In this case 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 must be -2c. So the probabilities obey the discretized diffusion equation:
which, when c is scaled appropriately and the P distribution is smooth enough to think of the system in a continuum limit becomes:
Which is the
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:
Since the sum of the squares of the amplitudes must be constant, must be unitary:
:or, in matrix notation,:
The rate of change of U is called the
HamiltonianH, up to a traditional factor of i:
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:
::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 which 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 is the time differential equation:
In the case that left and right are symmetric, c is real. By redefining the phase of the wavefunction in time, , 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.:which is the right choice of phase to take the continuum limit. When c is very large and psi is slowly varying so that the lattice can be thought of as a line, this becomes the free
If there is an additional term in the H matrix which is an extra phase rotation which varies from point to point, the continuum limit is the Schrödinger equation with a potential energy::
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 , 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::where the sum extends over all paths with the property that and . 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 balancewhen the stationary distribution has the property:
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 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.
When the R matrix obays detailed balance, the scale of the probabilities can be redefined using the stationary distribution so that they no longer sum to 1:
In the new coordinates, the R matrix is rescaled as follows:
and H is symmetric:
This matrix H defines a quantum mechanical system:
whose Hamiltonian has the same eigenvalues as those of the R matrix of the statistical system. The
Eingenvectorsare 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:
and t is allowed to take on complex values, the K' matrix is found by taking time imaginary.
For a quantum systems which is invariant under
time reversalthe 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 wavefunction, 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.
Applying the superposition principle to a quantum mechanical particle, the configurations of the particle are all positions, so the superpositions make a complex wave in space. The coefficients of the linear superposition are a wave which describes the particle as best as is possible, and whose amplitude interferes according to the
For any physical quantity in
quantum mechanics, there is a list of all the states where the quantity has some value. These states are necessarily perpendicular to each other using the Euclidean notion of perpendicularity which comes from sums-of-squares length, except that they also must not be i multiples of each other. This list of perpendicular states has an associated value which is the value of the physical quantity. The superposition principle guarantees that any state can be written as a combination of states of this form with complex coefficients.
Write each state with the value q of the physical quantity as a vector in some basis , a list of numbers at each value of n for the vector which has value q for the physical quantity. Now form the outer product of the vectors by multiplying all the vector components and add them with coefficients to make the matrix:where the sum extends over all possible values of q. This matrix is necessarily symmetric because it is formed from the orthogonal states, and has eigenvalues q. The matrix A is called the observable associated to the physical quantity. It has the property that the eigenvalues and eigenvectors determine the physical quantity and the states which have definite values for this quantity.
Every physical quantity has a
Hermitian linear operatorassociated to it, and the states where the value of this physical quantity is definite are the eigenstates of this linear operator. The linear combination of two or more eigenstates results in quantum superposition of two or more values of the quantity. If the quantity is measured, the value of the physical quantity will be random, with a probability equal to the square of the coefficient of the superposition in the linear combination. Immediately after the measurement, the state will be given by the eigenvector corresponding to the measured eigenvalue.
It is natural to ask why "real" (macroscopic, Newtonian) objects and events do not seem to display quantum mechanical features such as superposition. In 1935,
Erwin Schrödingerdevised a well-known thought experiment, now known as Schrödinger's cat, which highlighted the dissonance between quantum mechanics and Newtonian physics, where only one configuration occurs, although a configuration for a particle in Newtonian physics specifies both position and momentum.
In fact, quantum superposition results in many directly observable effects, such as
interferencepeaks from an electron wavein a double-slit experiment, although it can be shown that these effects are small for cats. The superpositions, however, persist at all scales, absent a mechanism for removing them. This mechanism can be philosophical as in the Copenhagen interpretation, or physical.
If the operators corresponding to two observables do not
commute, they have no simultaneous eigenstates and they obey an uncertainty principle. A state where one observable has a definite value corresponds to a superposition of many states for the other observable.
* A scheme of demonstration, similar to the two-slit experiment.
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