# Papers from the beginning of quantum mechanics

### From Institute for Theoretical Physics II / University of Erlangen-Nuremberg

Note: These papers are in German. English translations may be found in the book 'B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1'.

**Heisenberg's original matrix mechanics** - This is the work that created the modern theory of quantum mechanics
(Heisenberg 1925). Heisenberg wanted to tackle the question of how to predict correctly
the intensities of atomic transition lines, as Bohr had already clarified how to obtain the transition frequencies.
Heisenberg began by noticing that, according to Bohr, the correct quantum transition frequencies do not depend just
on the current state of motion (as do the frequencies of emitted radiation for a classical orbit), but rather on two states (initial and final).
Likewise, in classical theory, the intensities of emitted radiation would be given by the squares of the Fourier amplitudes of
the oscillating dipole moment for a given orbit. In an ingenious step, Heisenberg then postulated
that instead of a set of Fourier amplitudes for a given orbit (enumerated by one index), one would have to introduce a set
of amplitudes depending on two indices, one for the initial, the other for the final state. He assumed that the
equations of motion for those amplitudes looked formally the same as in classical theory (Heisenberg equations of motion).
The last crucial ingredient is the commutation relation. This he derived by looking at the linear response of an electron to
an external perturbation (essentially deriving something like Kubo's formula, containing the commutator) and then demanding that the short-time
response would be always that of a free, classical electron. This fixes the commutator between position and momentum.
Thus was born matrix mechanics.
He applied this immediately to the harmonic oscillator and also dealt with the anharmonic oscillator using perturbation theory.

See also Heisenberg's Nobel Lecture from 1933 to learn more about his view on these developments, and the slightly earlier overview (Heisenberg 1928 (Naturwissenschaften)) that also includes much of the developments before matrix mechanics.

**The three-men work (Dreimännerarbeit)** - The formalism of matrix mechanics was then developed fully by Born, Jordan and Heisenberg
(Born, Jordan and Heisenberg 1926). They discuss:
canonical transformations, perturbation theory, angular momentum, eigenvalues and eigenvectors. In addition to the
formalism, that work also contains the earliest discussion of a quantum field theory: A linear chain of masses coupled by springs
is quantized and solved by going over to normal modes. As a result, they find the Planck spectrum of thermal equilibrium, as a direct
consequence of the newly developed quantum mechanics!

**The hydrogen atom in matrix mechanics** - Wolfgang Pauli (Pauli 1926) managed to
apply the new matrix mechanics to the hydrogen atom. He found the correct energy spectrum, as well as the correct Stark effect corrections to the
energy in an applied electric field. In this solution, he makes use of the Runge-Lenz vector which is an additional conserved quantity known from
classical mechanics for the Kepler problem, denoting the orientation of the elliptical orbit in space.

**The Schroedinger equation** - Shortly after Heisenberg's work, Schroedinger came up with the equation that now carries
his name. The essential idea was to start from the Hamilton-Jacobi equation, claim the action is the logarithm of some wave function psi (think WKB!),
and derive a quadratic form of psi that is to be extremized (Schroedinger equation from the variatonal principle). This leads to the stationary Schroedinger equation, which he then solves
for the hydrogen atom, as well as for the harmonic oscillator, the rotor and the nuclear motion of the di-atomic molecule (Schroedinger 1926a and Schroedinger 1926b).

**The probabilistic interpretation** - While for a single electron inside an atom it might still be conceivable to view the wavefunction as some sort of smeared-out charge density, this view clearly becomes untenable when one moves to scattering processes, where the final scattered wave spreads out over a large region of space, whereas physically the particle will be detected at a point-like location. Thus it happened that during the investigation of quantum-mechanical scattering processes Max Born (Born 1926) was lead to the conclusion that the wave function has something to do with the probability of detecting a particle at some location. He at first incorrectly guessed the wave function itself gives the probability, but then inserted a famous footnote (on page 3 of this work) that says one should take the square! See also Max Born's Nobel lecture from 1954 for a discussion of the development of quantum mechanics.

**The uncertainty relation** - Heisenberg showed that the precision with which position and momentum can be measured cannot be arbitrarily high for both these quantities simultaneously (Heisenberg 1927). This work contains also the discussion of the famous "Heisenberg microscope" gedankenexperiment, where one tries to determine the position of an electron only to find that by doing so one destroys any interference pattern that might have existed without this act of observation.

**Spin** - The electron spin was introduced by Wolfgang Pauli (Pauli 1927) as
an additional discrete degree of freedom that could take two values. (Note: The Pauli spin matrices make their first appearance on page 8 of this work)

*Still missing: Fermions/bosons, beginnings of quantum field theory; foundational aspects (like Schroedinger's cat) should become a separate page*