Schroedinger Equation and the Probability Wave Function I

http://www.youtube.com/watch?v=6Q4_nl0ICao

The above youtube link shows the thought experiment described in the class and also available in the first chapter of Feynman Lectures III. It clearly tells us that the waves describing matter are actually probability waves such that the modulus squared of this probability wave amplitude describes the probability of the particles location and behaves just like the intensity of a normal ( linear) wave ( like an EM Wave).

This may be summarized as

For normal waves Intensity I ~ |amplitude|²

For probability waves Probability P ~ |probability wave amplitude|²

So we may describe the matter waves by a Probability Wave function/Wave Function
( considering only 1 space dimension x for simplicity)
Ψ(x,t)=A e^i(kx-ωt) where A is a constant. This wave function
is expected to satisfy a linear wave equation so that the superposition principle holds and must be consistent with the following relations.

p=ℏk; , E=ℏω=P²/2m + V(x)
or E=ℏ ²K²/2m + V(x)=ℏω

From these relations and the form of the wave function Ψ(x,t) it is obvious that
the equation must be second order in space derivatives ( for the k² and
first order in the time derivative ( for ω). So we can assume a form of the equation to be

[ α∂²/∂ x² + V(x)]ψ (x,t)=β(∂/∂ t) ψ(x,t)

Using the form of ψ(x,t) this gives -αk² +V =∓iβω this has two solutions β=± i ℏ. Assuming the + sign ( - is equivalent) we have the Schroedinger equation in 1 space dimension as

[ -ℏ²/2m ∂²/∂ x² + V(x)]ψ (x,t)=i ℏ(∂/∂ t) ψ(x,t)

This can easily be generalised to 3 dimensions as

[ -ℏ²/2m ∇² + V(x)]ψ (r ,t)=i ℏ(∂/∂ t) ψ(r ,t)

This is refered to as the Time Dependent Schoredinger Equation

It can be shown that the solution to this equation ψ is a complex valued function of space and time. Hence ψ is NOT MESURABLE.
The Probability density of location of the particle between x to x + dx is then proportional to P(x,t)~ ψ². P(x,T) must be real and positive semi definite that is either positive or zero.

The state of the system is given by the wave function ψ and knowing the state
at time t=0 the Schroedinger equation predicts the state at time t=t₀. However notice that since all Physics is contained in |ψ|² the knowledge of all physical characteristic of the system at t=0 does not determine the function ψ(x, 0).

Quoting Max Born "Motion of the particles in QM conforms to the laws of probability however probability itself is propagated in CAUSAL FASHION through the Schroedinger equation.

The total probability is given as the volume integral over all space ∫|psi;|² d x and it must be finite so that the integral must converge. This requires the wave function to be a SQUARE INTEGRABLE or an L² function. The total probability may be normalized to 1.

Notice now that the probabilities corresponding to two wave functions are individually P₁ and P₂ but when they are superposed their
probability
P₁₂=|psi;₁ +psi;₂|²=P₁ + P₂ + Cross Terms. The cross terms gives rise to the interference pattern just like the cross terms in the intensity of superposed EM waves like light in a Youngs Double slit experiment described at the beginning.