This Quantum World/Serious illnesses/Born

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Born[edit]

In the same year that Erwin Schrödinger published the equation that now bears his name, the nonrelativistic theory was completed by Max Born's insight that the Schrödinger wave function \psi(\mathbf{r},t) is actually nothing but a tool for calculating probabilities, and that the probability of detecting a particle "described by" \psi(\mathbf{r},t) in a region of space R is given by the volume integral



\int_R|\psi(t,\mathbf{r})|^2\,d^3r=\int_R\psi^*\psi\,d^3r


— provided that the appropriate measurement is made, in this case a test for the particle's presence in R. Since the probability of finding the particle somewhere (no matter where) has to be 1, only a square integrable function can "describe" a particle. This rules out \psi(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}, which is not square integrable. In other words, no particle can have a momentum so sharp as to be given by \hbar times a wave vector \mathbf{k}, rather than by a genuine probability distribution over different momenta.

Given a probability density function |\psi(x)|^2, we can define the expected value



\langle x\rangle=\int |\psi(x)|^2\,x\,dx=\int \psi^*\,x\,\psi\,dx


and the standard deviation  \Delta x = \sqrt{\int |\psi|^2(x-\langle x\rangle)^2}

as well as higher moments of |\psi(x)|^2. By the same token,



\langle k\rangle=\int \overline{\psi}\,^*\,k\,\overline{\psi}\,dk  and  \Delta k=\sqrt{\int |\overline{\psi}|^2(k-\langle k\rangle)^2}.

Here is another expression for \langle k\rangle:



\langle k\rangle=\int \psi^*(x)\left(-i\frac d{dx}\right)\psi(x)\,dx.

To check that the two expressions are in fact equal, we plug  \psi(x)=(2\pi)^{-1/2}\int \overline{\psi}(k)\,e^{ikx}dk  into the latter expression:



\langle k\rangle=\frac1{\sqrt{2\pi}}\int \psi^*(x)\left(-i\frac d{dx}\right)\int \overline{\psi}(k)\,e^{ikx}dk\,dx=\frac1{\sqrt{2\pi}}\int \psi^*(x)\int \overline{\psi}(k)\,k\,e^{ikx}dk\,dx.


Next we replace \psi^*(x) by (2\pi)^{-1/2}\int \overline{\psi}\,^*(k')\,e^{-ik'x}dk'  and shuffle the integrals with the mathematical nonchalance that is common in physics:



\langle k\rangle= \int\!\int \overline{\psi}\,^*(k')\,k\,\overline{\psi}(k) \left[\frac1{2\pi}\int e^{i(k-k')x}dx \right]dk\,dk'.


The expression in square brackets is a representation of Dirac's delta distribution \delta(k-k'), the defining characteristic of which is  \int_{-\infty}^{+\infty} f(x)\,\delta(x)\,dx = f(0)  for any continuous function f(x). (In case you didn't notice, this proves what was to be proved.)

Heisenberg[edit]

In the same annus mirabilis of quantum mechanics, 1926, Werner Heisenberg proved the so-called "uncertainty" relation


\Delta x\,\Delta p \geq \hbar/2.


Heisenberg spoke of Unschärfe, the literal translation of which is "fuzziness" rather than "uncertainty". Since the relation \Delta x\,\Delta k \geq 1/2 is a consequence of the fact that \psi(x) and \overline{\psi}(k) are related to each other via a Fourier transformation, we leave the proof to the mathematicians. The fuzziness relation for position and momentum follows via p=\hbar k. It says that the fuzziness of a position (as measured by \Delta x ) and the fuzziness of the corresponding momentum (as measured by \Delta p=\hbar\Delta k ) must be such that their product equals at least \hbar/2.