Time Independent Schrödinger's Equation

We want an analogous differential equation for quantum mechanics, which we can solve to find the form of the wave functions describing particles experiencing a force or potential gradient.

---- Schrödinger

Time Independent Schrödinger's Equation

Suppose a particle in energy eigenstate with quantum wave , potential energy and definite total energy , then we have the following equation hold: In one dimension, it deduces to

The Infinite Square Well

Def Infinite Square Well
Suppose we have a particle trapped in a potential whereThe solution to the infinite square well problem is simply:We call it ground state if , and all others excited states. And the number of zeros of aside from and is called nodes.
Prop The number of nodes is , is symmetric if is odd, antisymmetric if is even.
Prop Each solution to the infinite square well problem forms an energy eigenstate, the energy spanning from to is definitely given by .

Orthogonality of Energy Eigenstates

Def Kronecker Delta
For a set of normalized, mutually orthogonal eigenfunctions, we can thus write where is defined as Kronecker delta,

Prop The solutions to the time independent Schrödinger’s equation form a inner product space of eigenfunctions as orthonormal basis, where the inner product is defined as

Properties of Energy Eigenstates

Def Bound States
Bound states are states in the potential, that is . It is unbounded or scattering if .

Def Degeneracy
The two different energy eigenstates are degenerate if they yield same energy.

Proposition

There is no degeneracy for bound states in one-dimensional potentials.

Proof Define scaled energy and scaled potential respectively: Suppose there is such a degeneracy so that there are and , different from each other and both corresponding to the same energy , thus the same value of . We have that the following equations hold: Multiplying the top equation by and the bottom equation by and subtracting we get: It follows that , hence . The constant can be evaluated by examining the LHS for . We then have that and , since they are bound states, while the derivatives are bounded. It follows then that the LHS vanishes as and therefore. That is . Hence, This implies exists some constant :And thus Therefore we have shown that the wave functions and are equivalent, they indicate the same energy eigenstate.

Theorem

The energy eigenstates can be chosen to be real.

Proof Consider an eigenstate of the TISE given by . Then we have Now if is different from then it must be a degenerate solution. By superposition we can get two real degenerate solutions: $$\psi_{r}=\frac12(\psi+\psi^{}),\quad \psi_{im}=\frac1{2i}(\psi-\psi^{})$$$\square$

Corollary If dealing with a 1-D potential with bound states, any energy eigenstate is real up to a phase.
Proof By theorem, in a 1-D potential with bound states, we have Thus . Hence since is real, is up to a phase equal to a real solution.

Thrm If the potential is an even function of (i.e. ), the eigenstates can be chosen to be even or odd under reflection. Specially, if we are dealing with a 1-D symmetric potential, then any bound state must be even or odd.
Proof

The Finite Square Well

Def Finite Square well
Suppose we have a particle in a space with potential whereThe odd solutions are where . And the unique energies for the even case are given by