I was reading Shankar's Principles of Quantum Mechanics when on page 65, he starts talking about infinite spaces and operators in them. He introduces an operator $K$, which in the $x$ basis takes the form $K=-iD$ with $D = d/dx$ in this basis. He then solves the eigenvalue and vector problem for this operator and finds that in the $x$ basis, the eigenkets $|k \rangle$ take the form
$$\psi_k(x)= \langle x|k\rangle = Ae^{ikx}.$$
However, he later shows that such functions cannot be normalized, since
$$\langle k|k'\rangle = \int_{-\infty}^{\infty}\langle x|k' \rangle \langle k | x \rangle dx=\int_{-\infty}^{\infty}e^{i(k-k')x}dx=2\pi\delta(k-k').$$
My question then is, if $\psi_k(x)$ is not normalizable, then how can $|\psi_k(x)|^2$ represent a probability density? and what is the physical interpretation for $|\psi_k(x)|^2$ in these cases?

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    $\begingroup$ A plane wave doesn't represent a single particle (otherwise the probability to find that particle in any given volume would be zero) but it does represent the important physical case of a beam of particles with momentum k. In the classical case we aren't counting the number of particles but the number of particles that are passing an area unit per time unit, which would be proportional to an intensity. If we apply this to e.g. a step potential, then there are one incoming and two outgoing waves. By taking the ratios we calculate reflection and transmission coefficients. $\endgroup$ Apr 28 at 17:34
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    $\begingroup$ We simply give up the probability interpretation for a while because it is not normalisable. Instead, normalisable wavefunctions can still be expanded in terms of these eigenfunctions, and it is those normalisable wavefunctions that have the probability interpretation. $\endgroup$ Apr 29 at 5:43

1 Answer 1


Although an absolute notion of probability density $P(x)=|\psi(x)|^2$ does not make sense for a non-normalizable wavefunction $\psi(x)$, there is still a relative/comparative notion of probability density by considering ratios $|\psi(x_1)|^2~:~ |\psi(x_2)|^2$ between two positions $x_1$ and $x_2$.

This relative probability interpretation is utilized when studying scattering in QM, such as, reflection/transmission in 1D, cf. this related Phys.SE post.


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