Time Dependent Schrödinger's Equation
The general solution of TDSE is given by where are the solutions to the TDSE and . Given the initial wave function , we can obtain all the coefficients :
Proposition
Prop Linear superposition of any solutions to the TDSE is still a solution.
Lemma
Lemma Given any potential
, that is real and time independent, the separable solutions of the TDSE are valid. Proof If the potential only depends on position, we are able to write the TDSE as an equation where each side depends on different variables: So we can see that separable solutions satisfy the TDSE if the potential is time-independent.
Proposition
Prop All solutions of the time-independent Schrödinger’s equation have the time dependence of the form
where .
Proposition
Prop A wave function is normalized at time
then it is normalized for all time. Proof Consider the derivative of normalization: By TDSE, we have Substitute into the integral, we then get
Proposition
Prop The normalization condition for the wave function obtained by linear superposition is
Definition
Def Stationary State We say that a particle is in the stationary state if observable properties of the particle do not change with time.