Introduction: A Strange Normalization
For discrete bases, \(\langle m|n\rangle = \delta_{mn}\) (the Kronecker delta: 1 if equal, 0 otherwise). For continuous bases like \(|x\rangle\), we need the Dirac delta function \(\delta(x - x')\)—an infinitely sharp spike.
The Dirac Delta Function
The Dirac delta function \(\delta(x)\) is defined by its action under integration:
\[\int_{-\infty}^{\infty} f(x)\delta(x - a) \, dx = f(a)\]Intuitively: \(\delta(x) = 0\) for \(x \neq 0\), but "infinite" at \(x = 0\), with total area 1.
Normalization of Continuous Eigenstates
Position eigenstates are normalized as:
\[\langle x'|x\rangle = \delta(x - x')\]This is called delta-function normalization. Unlike discrete states, \(\langle x|x\rangle\) is not 1—it's "infinite"!
Worked Examples
Example 1: Sifting Property
Evaluate \(\int_{-\infty}^{\infty} x^2 \delta(x - 3) \, dx\):
The delta "sifts out" the value at \(x = 3\):
\[\int_{-\infty}^{\infty} x^2 \delta(x - 3) \, dx = 3^2 = 9\]Example 2: Consistency with Completeness
From \(\int |x\rangle\langle x| \, dx = \hat{I}\), apply to \(|x'\rangle\):
\[\int |x\rangle\langle x|x'\rangle \, dx = |x'\rangle\] \[\int |x\rangle \delta(x - x') \, dx = |x'\rangle \checkmark\]The delta function correctly picks out \(|x'\rangle\).
Example 3: Position Wavefunction of Position Eigenstate
What is the wavefunction of \(|x_0\rangle\)?
\[\langle x|x_0\rangle = \delta(x - x_0)\]A particle with definite position \(x_0\) has a wavefunction that's a spike at \(x_0\).
The Quantum Connection
The delta function appears throughout quantum mechanics:
- Normalization of plane waves: \(\langle p'|p\rangle = \delta(p - p')\)
- The delta-function potential: \(V(x) = -\alpha\delta(x)\)
- Commutation relation: \([\hat{x}, \hat{p}] = i\hbar \Rightarrow \langle x|\hat{p}|x'\rangle = -i\hbar\delta'(x-x')\)
It's the bridge between the abstract and the concrete in position space.