Use of The Wave Function In The Calculation of Expectation Values  

Any physical property moving in one dimension which does not depend upon time can be calculated from the time-independent (stationary-state) wave function of the system . In these calculations, the probability of finding the particle in any region of the one-dimensional space is calculated using Born’s postulate, according to which the probability .For example, the wave function helps us to calculate observables like position, momentum etc. However, if we carry out an experiment to measure the position x of the particle, may not give the result of that particular experiment. But if we repeat the experiment a number of times, the value of the position s calculated form is the same as the mean value of the different experiments. This mean (or average) value, denoted by <x> is called the expectation value of the observable x.

According to one of the postulates fo quantum mechanics, as operator for the position co-ordinate x is the multiplier x, therefore the expectation value will be given by
              
Where p(x) dx is the probability calculated by Born’s postulate.

Similarly, to calculate the expectation value of any other physical quantity (or observable),G, remembering that the operator for G is represented by and applying the basic postulates of quantum mechanics, the expectation value <G> will be given by
                                          
Which would be same as the mean value of the results of the repeated experiments for the measurement of G in the state.

It may be noted carefully that is not a simple factor like x. It is an operator the quantity G. Hencecannot be rearranged to give . Thus must be allowed to operate on . For example, according to one of the postulates fo quantum mechanism, for the momentum.

p(x) the operator. take derivative with respect to x and multiply the result by.

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