In quantum mechanics, the expectation value of a mechanical quantity is given by

where  is the operator for the mechanical quantity and ψ is the wave function for the state of the system. The integrals are over all permitted values of the coordinates of the system. The expectation value is defined as the prediction of the mean of a large number of measurements of the mechanical quantity, given that the system is in the state corresponding to ψ prior to each measurement.

For a particle moving in the x direction only and confined to a region on the x axis from x = 0 to x = a, the integrals are single integrals from 0 to a and pÌ‚_x is given by („/i)ˆ‚/ˆ‚x. The wave normalized function is

Normalization means that the integral in the denominator of the expectation value expression is equal to unity.

(a) Show that this wave function is normalized.
(b) Find the expectation value of x.
(c) The operator corresponding to

Find the expectation value of p_x.
(d) Find the expectation value of p²_x.

Transcribed Image Text:

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