In non-relavistic quantum mechanics, governed by the Schrodinger equation, the probability of finding a particle does not change with time.
Prove it, Begin with the time derivative of the total probability
Then use the Schrodinger equation to eliminate the partial time derivatives, integrate by parts, and show that the result is zero. Assume that the particle is well localised, so that are 0 when evaluated at .
(b) Does this procedure lead to the same conclusion if Wave function obeyKlein-Gordon rather than Shrodinger equation? Why and why not?
(a) The integrals from the two integrals by parts cancel each other, and the result is .
(b) It will not work if the wave function obeys Klein-Gordon equations.
The given integral is .
The symmetric wave function is given as,
The Anti-symmetric wave function is given as,
The time derivative of the probability density function is given, as shown below.
The integrals from the two integrals by parts, and we obtain:
The integrals from the two integrals by parts cancel each other, and the result is .
It will not work if the wave function obeys the Klein-Gordon equation.
Because the first-time derivative of the wave function does not appear in the equations.
So we cannot trade them into space derivatives.
Equation (12-7) assumes a matter-dominated universe in which the energy density of radiation is insignificant. This situation prevails today and has to do with the different rates at which the densities of matter and radiation vary with the size of the universe. Matter density is simply inversely proportional to the volume, obeying , where is the matter density now. Radiation density, however, would be proportional to (Not only does the volume increase, but also all wavelengths are stretched in proportion to R. lowering the energy density by the extra factor.) This density drops faster As the universe grows, but it also grows more quickly in the backward time direction. In other words, long ago, the universe would have been radiation dominated. Show that if the function used for matter density in equation (12-7) is replaced by one appropriate to radiation, but retaining the assumption that K' and are both 0, then the scale factor would grow as
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