The superposition principle is the idea that a system is in all possible states at the same time, until it is measured. After measurement it then falls to one of the basis states that form the superposition, thus destroying the original configuration. The superposition principle explains the "quantum weirdness" observed with many experiments. A classic example of this is the double-slit experiment. Here, two slits in a barrier allow for the passage of (for example) electrons. The result of this experiment is an interference pattern not predicted by classical mechanics.
The superposition principle states that a statefunction (Y) can be expanded as a linear combination of the normalized eigenstates (j_{n}) of a particular operator that constitute a basis of the space occupied by Y. For the discrete case:
where the coefficients b_{n} are, in general, functions of time and are given by:
QM Statefunction | 3-Space Vector | |
Vector | A(x, y, z) = xi + yj + zk | |
Basis vectors | Eigenstates: j_{n} | Unit vectors: i, j, k |
Projections | b_{1}, b_{2}, b_{3},... | x, y, z |
Physical meaning of projections | |b_{n}|^{2} = probability of finding the eigenvalue for the eigenstate j_{n} upon measurement. | Extension in space. |
Result of measurement | The state Y is destroyed. The system falls to one of the eigenstates j_{n} after measurement. | No change to A, i.e. all original components of A are intact. |
As an example, consider the expectation value of energy áEñ for a discrete system is in state Y. The normalized eigenfunctions of energy are j_{n} and the eigenvalues are E_{n}. The expectation (average) value is the sum:
where |b_{n}|^{2} is the absolute value of the coefficients of expansion that are given above. This is just the weighted average of the possible values of the observable E.
From the definition of the expectation value áEñ
Suppose you have an experiment like the one in the figure. Electrons are fired at the screen. Here, some kind of detector is behind the screen and records the impact (intensity) of the electrons. First slit one is closed, then slit two. The results are illustrated in (a) and (b) respectively. The intensity for the electron passing through the first slot is I_{1} and the intensity for the electron passing through the second is I_{2}. The result with both slits open is shown in (c), where an interference pattern is observed. Classical physics would predict an intensity that is merely the addition of the two individual intensities, or
This does not account for the interference pattern, however.
To deal with this pattern we consider the wavefunction (i.e., the quantum mechanical solution), Y, for which the intensity is given by,
So for the first slot only being opened, I_{1} = | Y_{1} | ^{2} and for the second only, I_{2} = | Y_{2} | ^{2}.
The resultant wavefunction for both slots being open is,
The superposition principle gives the resultant wavefunction for both slots being opened. Until a measurement is made, the system is "in" all possible states. Here, the possible states are the electron going through slot one (Y_{1}) and the electron going through slot two (Y_{2}).
The corresponding intensity is,
When the intensity I is expanded there results,
The 2 Re ( Y_{1}^{*} × Y_{2} ) term is called the "interference term." This results in the oscillation pattern in (c).
The superposition of states thus explains the quantum interference pattern. When both slits are open, the description of the system is the superposition of the states when each slot is opened individually (i.e., Y = Y_{1} + Y_{2}) and it is just this superposition that accounts for the interference. This is true until one tries to determine which path is taken by an electron, after which the state of the system collapses. The classical interpretation of particles bombarding a detector fails to adequately describe the situation.
A typical problem for the superposition principle is like this:
Quantum Superposition Another description of the superposition principle.
Quantum Superposition 2 And another description of the superposition principle.
Schrödinger's Cat Brief description of the Schrödinger's Cat problem and how the superposition principle applies to it.
Applet A superposition applet for a 2-dimensional box.
Wave Equations, Wavepackets and Superposition Explanation from UVA.
Double Slit Experiment Simulation of the double slit experiment.
References
Liboff, R. L. Introductory Quantum Mechanics, Fourth Edition. San Francisco, CA: Addison Wesley, 2003.
Schwabl, F.; Quantum Mechanics Second Revised Edition. New York, NY: Springer-Verlag, 1995.
Peleg, Y.; Pnini, Reuven; Zaarur, Elyahu Schaum's Outline and Theory and Problems of Quantum Mechanics. New York, NY: McGraw-HIll, 1998.
"Fundamentals of quantum information." From physicsweb.org: http://physicsweb.org/article/world/11/3/9.
* The superposition principle is often considered to be a postulate, although Liboff (4^{th} edition) does not explicitly say so. See "Postulates" for more information.