Normalization and inner products

Why normalization matters

If $|psi(x,t)|^2$ is probability density, then the total probability of finding the particle somewhere must be 1. For a one-dimensional particle on the full line, this means

$int_{-infty}^{infty}|psi(x,t)|^2dx=1$

A wavefunction satisfying this condition is normalized. If a wavefunction has the correct shape but wrong overall scale, it can often be normalized by multiplying by a constant.

Normalization constant

Suppose

$psi(x)=A f(x)$

where $A$ is a constant. The normalization condition gives

$1=int |A f(x)|^2dx=|A|^2int |f(x)|^2dx$

so

|A|=rac{1}{sqrt{int |f(x)|^2dx}}

The overall phase of $A$ usually has no physical effect for a single state, so $A$ is often chosen real and positive.

Square-integrable states

A physically normalizable bound-state wavefunction must be square integrable:

$int |psi(x)|^2dx<infty$

Plane waves like $e^{ikx}$ extend infinitely and are not normalizable in the usual sense on the full line. They are useful idealizations and can be treated with box normalization or Dirac delta normalization.

Inner products

The inner product of two wavefunctions is

angle=int_{-infty}^{infty}phi^*(x)psi(x),dx$$ This generalizes the dot product of vectors. The norm of a state is $$langle psi|psi angle=int |psi(x)|^2dx$$ A normalized state satisfies $$langle psi|psi angle=1$$ ## Orthogonality Two states are orthogonal if their inner product is zero: $$langle phi|psi angle=0$$ Orthogonal states represent mutually exclusive alternatives in many measurement contexts. For example, different bound-state energy eigenfunctions of the same Hermitian Hamiltonian are orthogonal if they have distinct energies. ## Expansion in basis states If ${u_n(x)}$ is an orthonormal basis, then a state can be expanded as $$psi(x)=sum_n c_n u_n(x)$$ where coefficients are found by projection: $$c_n=langle u_n|psi angle$$ If the state is normalized, then $$sum_n |c_n|^2=1$$ The quantity $|c_n|^2$ is the probability of obtaining the corresponding basis outcome, such as an energy measurement yielding $E_n$ when $u_n$ is an energy eigenstate. ## Hermitian operators Observables are represented by Hermitian operators. Hermitian operators have real eigenvalues and orthogonal eigenfunctions for distinct eigenvalues. The Hermiticity condition can be written as $$langle phi|hat{A}psi angle=langle hat{A}phi|psi angle$$ under suitable boundary conditions. ## The big idea Normalization ensures total probability equals one. Inner products measure overlap between quantum states, define norms, express orthogonality, and provide expansion coefficients. These ideas turn wave mechanics into vector geometry in an abstract state space.