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# 2.4 Operators and superposition

In quantum mechanics, an operator is a mathematical entity which converts one function into another function and is written with a ‘hat’on, for example , (pronounced ‘A hat’). Given an operator , the eigenvalue equation for that operator is

Equation label:(6)

Here, is a constant known as an eigenvalue, which may be complex, and is function known as an eigenfunction. There may be more than one eigenvalue and corresponding eigenfunction associated with each eigenvalue equation. The eigenvalue matrix equation, Equation (2) as described in Section 2.2 is an example of this type of equation with the operator written as a matrix and the eigenfunction as a column vector.

Consider an operator with two eigenvectors and two corresponding eigenvalues so that

Since and are both eigenfunctions or solutions of the eigenvalue equation any linear combination of and is also a function on which the operator can act. Such a linear combination, known as a superposition, is

where and are complex numbers.

In general, this superposition is not itself an eigen function of unless

## Exercise 7

The wave function of a free particle in quantum mechanics may be written as

Confirm that is an eigenfunction of (where indicates a partial derivative) and show that the eigenvalue is the energy of the free particle, . ( is the reduced Planck’s constant.)

### Answer

Operating on with we find that

Thus the free-particle wave function is an eigenfunction of and the corresponding eigenvalue is the energy, , associated with this wave.

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