Wavefunction Normalization

A wavefunction is a representation of a quantum state in a chosen basis. The default convention is to normalize square-integrable wavefunctions so that the total Born probability is one.

In one spatial dimension,

$$\int_{-\infty}^{\infty}\lvert\psi(x)\rvert^2\,dx=1.$$

In three spatial dimensions,

$$\int_{\mathbb R^3}\lvert\psi(\mathbf r)\rvert^2\,d^3r=1.$$

Probability Densities

The position-space probability density is

$$\rho(x)=\lvert\psi(x)\rvert^2$$

in one dimension, and

$$\rho(\mathbf r)=\lvert\psi(\mathbf r)\rvert^2$$

in three dimensions. A density is not itself a probability at a point. Probabilities are assigned to regions:

$$P(a<X<b)=\int_a^b \lvert\psi(x)\rvert^2\,dx.$$

The dimensions of $\psi$ depend on the coordinate measure. In one dimension, a normalized position-space wavefunction has units of length$^{-1/2}$ when $x$ has units of length.

Momentum-Space Normalization

With the default Fourier convention, normalization is preserved:

$$\int_{-\infty}^{\infty}\lvert\psi(x)\rvert^2\,dx = \int_{-\infty}^{\infty}\lvert\phi(p)\rvert^2\,dp =1.$$

The momentum-space density is $\lvert\phi(p)\rvert^2$. If a page uses a wave-number amplitude instead of a momentum amplitude, the Jacobian between $p$ and $k$ must be included.

Discrete and Continuous Bases

For a discrete orthonormal basis,

$$\lvert\psi\rangle=\sum_n c_n\lvert n\rangle, \qquad \sum_n \lvert c_n\rvert^2=1.$$

For continuous bases, normalizable wavefunctions are square-integrable. Formal objects such as plane waves and position eigenkets are delta-normalized distributions, not normalizable Hilbert-space vectors. Scattering pages may use delta normalization or box normalization, but they must declare it.

Common Mistakes

• Normalizing $\psi$ instead of $\lvert\psi\rvert^2$.
• Forgetting that $\psi(x)$ has dimensions.
• Confusing probability density with probability.
• Treating plane waves as normalizable states without declaring a distributional or box-normalized convention.
• Mixing momentum and wave-number normalization without the Jacobian.
• Renormalizing after every algebraic step without checking whether the step was unitary.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1980.

Exercises

1. Normalize $\psi(x)=A e^{-\alpha\lvert x\rvert}$ on the real line, with $\alpha>0$.

Solution

Require

$$1=\int_{-\infty}^{\infty}\lvert A\rvert^2 e^{-2\alpha\lvert x\rvert}\,dx.$$

The integral is

$$2\int_0^\infty e^{-2\alpha x}\,dx =\frac{1}{\alpha}.$$

Thus $\lvert A\rvert^2/\alpha=1$, so a real positive choice is $A=\sqrt{\alpha}$.

2. If $\psi(x)$ is normalized, what is the probability of finding the particle in $[a,b]$?

Solution

The probability is

$$P(a<X<b)=\int_a^b\lvert\psi(x)\rvert^2\,dx,$$

with endpoint conventions irrelevant for ordinary continuous densities.