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 cap a hat , (pronounced ‘A hat’). Given an operator , the eigenvalue equation for that operator is

cap a hat times f of x equals lamda times f of x

Equation label:(6)

Here, lamda is a constant known as an eigenvalue, which may be complex, and f of x 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 cap a hat with two eigenvectors and two corresponding eigenvalues so that

cap a hat times f sub one of x equals lamda sub one times f sub one of x and cap a hat times f sub two of x equals lamda sub two times f sub two of x full stop

Since f sub one of x and f sub two of x are both eigenfunctions or solutions of the eigenvalue equation any linear combination of and is also a solution. Such a linear combination, known as a superposition, is

f of x equals a sub one times f sub one of x plus a sub two times f sub two of x

where a sub one and a sub two are complex numbers.

Exercise 7

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

normal cap psi sub free of x comma t equals cap a times e super i left parenthesis k times x minus omega times t right parenthesis

Confirm that normal cap psi sub free of x comma t is an eigenfunction of i italic h over two pi times prefix partial differential of solidus prefix partial differential of of t (where indicates a partial derivative) and show that the eigenvalue is the energy of the free particle, italic h over two pi times omega . ( is the reduced Planck’s constant.)

Answer

Operating on normal cap psi sub free of x comma t with i italic h over two pi times prefix partial differential of solidus prefix partial differential of of t we find that

multiline equation row 1 i italic h over two pi times prefix partial differential of divided by prefix partial differential of of t times normal cap psi sub free of x comma t equals i italic h over two pi times prefix partial differential of divided by prefix partial differential of of t times left parenthesis cap a times e super i left parenthesis k times x minus omega times t right parenthesis right parenthesis row 2 equals negative italic h over two pi times omega times cap a times i super two times e super i left parenthesis k times x minus omega times t right parenthesis row 3 equals italic h over two pi times omega times normal cap psi sub free of x comma t

Thus the free-particle wave function normal cap psi sub free of x comma t equals cap a times e super i left parenthesis k times x minus omega times t right parenthesis is an eigenfunction of i italic h over two pi times prefix partial differential of solidus prefix partial differential of of t and the corresponding eigenvalue is the energy, italic h over two pi times omega , associated with this wave.

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