Superposition of states is a fundamental concept in quantum
mechanics. It is used in many different situations and is the
principle behind the paradox of Schrodinger's Cat. As quantum
mechanics became developed it was found that quantum superposition was
a necessary and unavoidable aspect of quantum mechanics, underscoring
the probabilistic nature of all quantum mechanical equations.

### High-level overview

Quantum mechanical systems are described in such a
way that their states can be combined, or *superposed*. This
superposition means that the system is effectively in both states at
the same time, and thus each component state evolves as it would as an
independent state. When the state of the system is observed, it
resolves into one of the states in the superposition. This is called
the 'collapse of the wavefunction', and is very poorly
understood.

If a system in a superposition of states interacts with another
quantum system, that system also enters a superposition of states. If
the other system is already in a superposition of states then more
possibilities are added to each system. This is the idea behind
Schrodinger's Cat; a random quantum system such as a radioactive atom
enters a superposition (decay or no decay) and interacts with a
detector putting it in a superposition (detection or no detection), and
so on until the cat is put in a superposition (dead or alive).

Schrodinger's cat highlights a philosophical problem in quantum
mechanics: what can collapse a wavefunction? Do we need a sentient
scientist to peek in the box for the wavefunction to collapse? Can the
cat collapse the wavefunction? Can the particle detector? Or do we,
too, enter a superposition of states when we look in the box? (This is
effectively the Many-Worlds Interpretation of quantum mechanics) In
practice, physicists often consider that the wavefunction collapses
whenever the state the system is in becomes important, such as a
particle interaction or interaction with a large, classical system.

### Mathematical basis

The principle of superposition is an important part of many
classical theories, especially those that involve wave
phenomena. In quantum mechanics, it takes a new and unusual meaning
based on probability. Most ordinary quantum mechanical systems are described by wavefunctions that are solutions of the Schrodinger Equation. Since the Schroedinger equation is linear, any linear combination of solutions is itself a solution. If we have the solutions ψ_{i}, then the general solution is given by

Ψ = ∑a_{i}ψ_{i}

Here, the probability that the system will be observed in state ψ

_{i} is |a

_{i}|

^{2}. This is the mathematical statement of a superposition of states.

This principle can be generalised to quantum systems not described by the Schrodinger Equation, such as spin. In this case a state is denoted by the Dirac ket |ψ_{i}>, and thus the superposition is denoted by

|Ψ> = ∑a_{i}|ψ_{i}>

Otherwise, the formalism is the same as for Schrodinger wavefunctions.

### The Necessity of Superposition

At this point, superposition seems like an interesting tool but it
does not seem like a fundamental aspect of quantum mechanics. It
arises naturally through operator theory. Every physical observable,
such as position or momentum, has a corresponding operator which acts
upon quantum states, producing a new state function. There are also
operators that correspond to symmetry transformations.

Each operator has a set of states where, when the operator is
applied to the state, it produces a constant multiple of the original
state. These states are called eigenstates and the constant that
multiplies it is called the eigenvalue. Symbolically, we have:

[Q]|ψ> = Q|ψ>

where [Q] is an arbitrary operator, |ψ> is the
eigenstate vector, and Q is the eigenvalue. Here, the eigenvalue Q is
the value of the observable for the state |ψ>, and it is an
exact, definite value. This is the physical meaning of an eigenstate;
it is a state where the value of the observable is definite and
without uncertainty.

Operators that correspond to physical quantities are Hermitian,
which, among other things, means that the set of eigenstates of the
operator is *complete*, meaning that any state can be expressed
as a linear combination of the eigenstates. That is, the eigenstates
can be superposed to form any state whatsoever.

Where this becomes sticky is when you consider multiple
operators. A pair of operators may or may not commute, depending on
their properties. A basic theorem in quantum mechanics shows that
two commuting operators have a common set of eigenstates. However,
when two operators do not commute, such as the position and momentum
operators, they cannot have the same set of eigenstates.

Since the sets of eigenstates are complete, the eigenstates of one
operator can be expressed as superpositions of the eigenstates of the
other operator. Whenever an observable is measured, the state
collapses into an eigenstate of the corresponding operator. It thus
naturally enters a superposition of the eigenstates of all operators
that don't commute with that operator. This has deep connections to the
Heisenberg Uncertainty Principle.

### Conclusions

Quantum superposition is at the root of a number of well-known
quantum systems, from the somewhat whimsical Schrodinger's Cat to
the common case of the Heisenberg Uncertainty Principle and more
obscure cases such as the kaon, neutrino oscillation, and CKM
mixing. It is also the principle that lends power to the concept of
quantum computing. Despite this, it is one of the most
counter-intuitive elements of quantum theory and is responsible for
many of the weird features of quantum systems.

**(CC)**
This writeup is copyright 2004 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.0/ .