If weintegrate if over some intervalinx
∫ba∂P(x,t)
∂tdx=−∫ba∂j(x,t)
∂x∂
∂t∫baP(x,t)dx=j(x=a,t)−j(x=b,t)the equation says that the rate of change of probability in an interval is equal to the probability flux
into the integral minus the flux out.
Extending this analysis to3 dimensions,
∂P(x,t)
∂t+∇·~ ~j(~r,t) = 0with
~j(~r,t) = ̄h
2 mi[
ψ∗(~r,t)∇~ψ(~r,t)−ψ(~r,t)∇~ψ∗(~r,t)]
7.3 The Schr ̈odinger Wave Equation
The normal equation we get, for waves on a string or on water, relates the second space derivative
to the second time derivative. The Schr ̈odinger equation usesonly the first time derivative,
however, the addition of theirelates the real part of the wave function to the imaginary part, in
effect shifting the phase by 90 degrees as the 2nd derivative would do.
− ̄h^2
2 m∂^2 ψ(x,t)
∂x^2+V(x)ψ(x,t) =i ̄h∂ψ(x,t)
∂tThe Schr ̈odinger equation isbuilt for complex wave functions.
When Dirac tried to make a relativistic version of the equation, wherethe energy relation is a bit
more complicated, he discovered new physics.
Gasiorowicz Chapter 3
Griffiths Chapter 1
Cohen-Tannoudji et al. Chapter
7.4 The Time Independent Schr ̈odinger Equation
Second order differential equations, like the Schr ̈odinger Equation, can be solved byseparation of
variables. These separated solutions can then be used to solve the problem ingeneral.