Q. Derive Schrodinger's time dependent wave equation. What is the physical significance of wave function used in this equation?
Ans.
Time Dependent Schrodinger's Equation : Let the wave function associated with a particle be represented where is the position vector and t is the time.
Now let us suppose that a particle is moving in +X direction then according to De-Broglie, there must be a wave associated with the moving particle. The wave function y, associated with the moving particle is given by
Physical interpretation of wave function : The quantity whose variation make up matter waves is called wave function. Normally wave function is represented by the symbol y. The wave function y, associated with a moving particle at the particular point x, y, z in space at the time t, gives the probability of finding the particle there at that time. The probability that something be in a certain place at a given time must lie between 0 and 1. An intermediate probability, say 0.2, means that there is a 20 % chance of finding the object. Now wave function y gives the amplitude of the wave and as we know that amplitude of any wave can either be positive or negative and a negative probability is meaningless. Hence y by itself cannot be an observable quantity or we can say that y has no direct physical significance.
There are certain basic postulates and essential definitions which are of fundamental importance.
1. Wave function y is known as quantum mechanical state of the system as it describes a physical system completely.
2. The intensity of a wave motion is proportional to the square of amplitude, so the intensity of the matter waves can also be defined by the square of its amplitude
3. The wave function y can be expressed by a complex function. In that case it's complex conjugate is given by y and it is obtained by replacing each i by i.
4. The probability of finding the particle in the interval dx around the point x at tim is dx. Similarly the probability of finding the particle in a volume element
5. Normalization condition. This is called normalization condition. The wave function which satisfies above equation is said to be normalized to unity.
6. Orthogonal wave function. Then these functions are said to be mutually orthogonal. This is known as orthogonality condition.
7. The expectation value or average value of the physical observable quantity say Q is Defined as
8. A wave function y must satisfy the following condition - wave function y and its first order derivative i.e. and must be finite, continuous, single valued everywhere and zero at large distances.