The probability of finding a particle between a and b at time t, is,…

Since the particle must be somewhere the statistical interpretation of QM requires the normalization condition,…

In general the average value of some function f() is (called the ‘expectation value’ in QM),…

,….. where p(j) is the probability of f(j).

So, for example, the expectation value for position, x, would be,…

Observables in quantum mechanics are represented by Hermitian operators, self-adjoint operators or matricies that are equal to their own transpose-conjugate. (i.e. position, momentum, Hamiltonian, etc). This is required so that eiginvalues are Real numbers, and to facilitate the association of these operators with the orthonormal basis underlying Hilbert space. On account of this, the x-operator above, can be moved over to operate on the second factor of the inner product…

Now, this position expectation value changes as the wavefunction evolves in time. The expectation value of this rate of change, the “velocity”, would be,…

, ….. or via Leibniz rule ,….

………………..(1)

The SchrÃ¶dinger equation shows what the rate of change of with respect to time must be,…

… and its complex conjugate,…

Substituting these into equation (1) above, allows the potential terms, , to cancel out, and results in,….

Factoring out a partial derivative operator and moving the constants outside the integral,…

………………..(2)

This equation can be simplified using ‘integration by parts’ which works as follows;

———————————————————-

Starting with a derivative of a product and expanding via Leibniz rule, ….

…. or,….

Integrating each side,….

The wavefunction will be zero at the boundary or at infinity, so that term can be dropped.

The result is equivalent to moving the f into the derivative and the g out, and then negating.

———————————————————-

Performing an ‘integration by parts’ on equation (2) above will move the into the , and the out, plus negating results in,…

Performing a integration by parts again, on the second term in parentheses, will move the into the partial derivative, and the out, and negating,….

,….

,….

,….

Now, momentum is mass times velocity, so multiplying both sides by m, will result in the ‘expectation value’ of momentum,….

This is similar in form to the expectation value equation for position above,….

Where the position operator, x, was placed between and . Therefore, momentum in quantum mechanics is associated with an operator,…

The first term on the right side of the SchrÃ¶dinger equation is the momentum form of kinetic energy….

kinetic energy

…

-mjg