Wave Function
Introduction:
In quantum mechanics, because of the wave-particle duality, the properties of the particle can be described as a wave. Therefore, its quantum state can be represented as a wave of arbitrary shape and extending over all of space. This is called a wave function. The wave function is usually complex and is represented by Ψ. Since the wave function is complex, its direct measurement in any physical experiment is not possible. It is just mathematical function of x, t etc. Once the wave function corresponding to a system is known, the state of the system can be determined. The physical state of system is completely characterized by a wave function.
Physical significance of a wave function:
The wave function contains information about the system it represents. Even though the wave function itself is not directly an observable quantity, the square of the absolute value of the wave function gives the probability of finding the particle at a given space and time. This probabilistic interpretation of wave function was given by Max Born in 1926. If Ψ is the wave function associated with a particle, the |Ψ|2 is the probability per unit volume that the particle will be found at the given point. The probability density is given by
where Ψ∗ is the complex conjugate of Ψ. For a particle restricted to move only long x− axis, the probability of finding it between x1 and x2 is given by
Since the probability of finding a particle any where in a given voluve must be one, we have
This condition is know as normalization.
Properties of a wave function:
A wave function has the following characteristics.
1. Ψ must be continuous and single-valued everywhere.
2. ∂Ψ/∂x, ∂Ψ/∂y and ∂Ψ/∂z must be continuous and single-valued everywhere.
3. Ψ must be normalizable.