# This Quantum World/Serious illnesses/Born

## Born

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 ${\displaystyle \psi (\mathbf {r} ,t)}$ is actually nothing but a tool for calculating probabilities, and that the probability of detecting a particle "described by" ${\displaystyle \psi (\mathbf {r} ,t)}$ in a region of space ${\displaystyle R}$ is given by the volume integral

${\displaystyle \int _{R}|\psi (t,\mathbf {r} )|^{2}\,d^{3}r=\int _{R}\psi ^{*}\psi \,d^{3}r}$

— provided that the appropriate measurement is made, in this case a test for the particle's presence in ${\displaystyle R}$. 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 ${\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} },}$ which is not square integrable. In other words, no particle can have a momentum so sharp as to be given by ${\displaystyle \hbar }$ times a wave vector ${\displaystyle \mathbf {k} }$, rather than by a genuine probability distribution over different momenta.

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

${\displaystyle \langle x\rangle =\int |\psi (x)|^{2}\,x\,dx=\int \psi ^{*}\,x\,\psi \,dx}$

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

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

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

Here is another expression for ${\displaystyle \langle k\rangle :}$

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

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

${\displaystyle \langle k\rangle ={\frac {1}{\sqrt {2\pi }}}\int \psi ^{*}(x)\left(-i{\frac {d}{dx}}\right)\int {\overline {\psi }}(k)\,e^{ikx}dk\,dx={\frac {1}{\sqrt {2\pi }}}\int \psi ^{*}(x)\int {\overline {\psi }}(k)\,k\,e^{ikx}dk\,dx.}$

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

${\displaystyle \langle k\rangle =\int \!\int {\overline {\psi }}\,^{*}(k')\,k\,{\overline {\psi }}(k)\left[{\frac {1}{2\pi }}\int e^{i(k-k')x}dx\right]dk\,dk'.}$

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

## Heisenberg

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

${\displaystyle \Delta x\,\Delta p\geq \hbar /2.}$

Heisenberg spoke of Unschärfe, the literal translation of which is "fuzziness" rather than "uncertainty". Since the relation ${\displaystyle \Delta x\,\Delta k\geq 1/2}$ is a consequence of the fact that ${\displaystyle \psi (x)}$ and ${\displaystyle {\overline {\psi }}(k)}$ 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 ${\displaystyle p=\hbar k}$. It says that the fuzziness of a position (as measured by ${\displaystyle \Delta x}$ ) and the fuzziness of the corresponding momentum (as measured by ${\displaystyle \Delta p=\hbar \Delta k}$ ) must be such that their product equals at least ${\displaystyle \hbar /2.}$