Lecture: Quantum mechanics. Classical physics

Quantum mechanics.

In 1900 Max Planck solved the paradox of black-body radiation by proposing that the exchange of energy between matter and radiation could take place only discontinuously, through radiation packets (quants) of a restricted size:

where

is the famous Planck’s constant, and is the frequency of the radiation exchanged. Planck took a conservative view of the quantum hypothesis: discontinuity might be a feature of energy exchange between matter and field but the exchanged packets of energy, the quanta, need not retain their identity during transmission through space.

In 1905 Einstein solved another problem of classical physics by proposing an even more radical quantum hypothesis: that there were free light quanta, ‘photons’, which did retain their identities while traveling through space. This idea enabled him to explain the classically puzzling photoelectric effect.

Experiments of Ernest Rutherford showed that atoms consisted of a small positively charged central core and an outer shell of negatively charged electrons. The puzzle for classical physics was in following: If the electrons are stationary then Maxwellian electromagnetism tells us that they will be attracted to the central core. If they rotate they should, as charged particles accelerated in an electric field, radiate their energy and very quickly collapse into the core.

Bohr’s solution to the puzzle was like Planck’s for black-body radiation: the electrons in an atom are to be allowed to be stable only in certain classically possible orbits, in those orbits in which the angular momentum of the electron is an integral multiple of Planck’s constant h. Classically the electron should go spiraling down towards the nucleus very rapidly. But Bohr simply forbade this.

From this idea, which fixed the energy levels of the electron in the hydrogen atom, together with the idea that the wavelengths of radiation emitted and absorbed by the atom were due to the transition between the allowed stable orbits, the spectrum of hydrogen could be accurately inferred. This was the great triumph for Bohr and the old quantum theory.

Bohr invented new scientific method: the correspondence principle. It tells us that the numbers given out by a classical and a quantum mechanical explanation of some phenomenon will converge as the quantum numbers get larger, as in other words the phenomenon gets less specifically quantum mechanical.

Modern quantum mechanics appeared in two mathematically very different forms in the years 1925 and 1926. Heisenberg developed matrix mechanics from a mathematical formulation of the correspondence principle. Schrodinger created wave mechanics under the influence of de Broglie’s matter/wave idea.

The French physicist Louis de Broglie proposed that matter is wavelike, thus inverting Einstein’s assertion that light is corpuscular. De Broglie’s theory was relativistic. Every particle, every electron, proton and so on was to have an associated wave, with frequency given by:

where m is the rest mass of the particle. When moving, a particle with momentum p has a de Broglie wavelength such that:

De Broglie explained why the electron in the Bohr hydrogen atom orbited in stationary states. Only in the stationary states does the de Broglie wave of the electron resonate and not self-interfere.

The two theories, of Heisenberg and Schrodinger appeared to be very different mathematically. In 1926 Schrodinger proved the two theories empirically equivalent.

In Schrodinger picture the state of a particle is given by so call wave function:

It is complex function and cannot be measured by any device. But through this function can be obtained the probability P that particle at time t has coordinate x :

The evolution of wave function is deterministic and follows Schrodinger equation:

If this equation has two solutions, then the linear combinations of these solutions also will be solution of equation.

This is called principle of superposition. The complete solution of Schrodinger equation is the superposition of all possible solutions. For example if particle can be in two states: 1) “alive”; 2) “dead” , then the complete description of particle’s state consists from superposition of these two states. Particle simultaneously “alive” and “dead”.

Dynamical variables in quantum mechanics defined as operators. For example, momentum has a form:

so momentum is operator of differentiation. It acts on a wave function.

The formalism of quantum mechanics describes microphysical systems:

1. 
   by assigning to them wave-function , and states can be added or superposed to give new states; and
   
2. 
   by assigning to their dynamical variables (position, momentum, angular momentum, energy) certain corresponding operators on wave-function such that
   
3. 
   from a knowledge of a system’s wave-function and the operator corresponding to any observable one can calculate the probability that a measurement of that observable on the system will have its result in any range we care to choose.
   

So quantum mechanics is not deterministic theory, it’s outcome is only probability.

Position operator, let us denote it as , acts on a wave-function by multiplication operator:

Momentum operator acts on a wave-function by differentiation:

These two operations do not commute with each other:

From quantum mechanical formalism follows that these two physical quantities cannot be observed simultaneously. They obey so called Heisenberg’s principle of uncertainty :

Seminar topics:

1. 
   Quantum mechanics
   
2. 
   Paradoxes of quantum mechanics
   
3. 
   Wave-particle dualism
   
4. 
   Louis de Broglie
   
5. 
   Niels Bohr
   
6. 
   Erwin Schrodinger
   
7. 
   Heisenberg
   
8. 
   Uncertainty Principle
   
9. 
   Schrodinger’s Cat