Superposition Principle

The Superposition Principle*

Introduction

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.

Discussion

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:

which is the projection of Y onto the eigenvector j_n. This representation is similar to how a 3-dimensional vector is a superposition of its projections onto the basis vectors i, j, k (which correspond to the eigenstates j_n). Unlike the "normal" vector, however, measurement of an observable will destroy the original statefunction. See Table 1 for a comparison of the two.

	QM Statefunction	3-Space Vector
Vector		A(x, y, z) = xi + yj + zk
Basis vectors	Eigenstates: j_n	Unit vectors: i, j, k
Projections	b₁, b₂, b₃,...	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.

Table 1: Comparison of Statefunction and Vector

Because the coefficients of expansion represent probabilities of what measurement will obtain, the superposition principle allows calculation of:

The expectation (average) value of an observable.
The probability that measurement of an observable will give a particular value.

Expectation Value

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.

Probability of Measurement

From the definition of the expectation value áEñ

we get,

or the probability of obtaining E_n, when the energy of the system (in the state Y ) is measured, is equal to |b_n|². Note that since the eigenstates j_n are normalized,

In other words, we are guaranteed to get an eigenvalue corresponding to some j_n.

The Double-Slit Experiment

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 I₁ and the intensity for the electron passing through the second is I₂. 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

I = I₁ + I₂

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,

I = | Y | ².

So for the first slot only being opened, I₁ = | Y₁ | ² and for the second only, I₂ = | Y₂ | ².

The resultant wavefunction for both slots being open is,

Y = Y₁ + Y₂

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 (Y₁) and the electron going through slot two (Y₂).

The corresponding intensity is,

I = | Y₁ + Y₂ | ²

When the intensity I is expanded there results,

I = | Y₁ + Y₂ | ² = I₁ + I₂ + 2 Re ( Y₁^* × Y₂ )

The 2 Re ( Y₁^* × Y₂ ) 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 = Y₁ + Y₂) 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.

Problem Solving

A typical problem for the superposition principle is like this:

The system is in the state Y, which is not necessarily an eigenstate of the observable to be measured.
We want to determine the possible outcomes of measuring an observable (say, A) with eigenfunctions j_n.
Expand the state Y in a superposition state of A. That is, represent the system as a linear combination of the b_nj_n terms with b_n = áj_n | Y ñ. Until measurement, the system is a superposition of all possible states.
The coefficients b_n of the expansion can then be used to determine the probability a particular result (i.e., eigenvalue) will be obtained from measurement.
After measurement, the system will then be in an eigenstate of A. The original state Y is destroyed. If another measurement is to be made, the new state (the eigenstate of A, j_n ) can be expanded as Y was and the procedure repeated.

See Example Problem

Links

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.