Chapter Two

Summary Development of Quantum Theory

Here is a summary revision of the development of Quantum Theory from its inception so that the reader may know the steps in the evolution of the theory. I have avoided all rigor and derivations. Interested readers can check any textbook for the details. This is intended only as an outline. It is not necessary that one understand all the mathematical formalism to take up the conclusions. Do not be disheartened if you do not understand the equations or derivations.

2.1 Discovery of Electron

In April 30, 1897, Joseph John Thomson announced that cathode rays were negatively charged particles, which he called 'corpuscles’. Thomson proposed the existence of elementary charged particles, which is now called electrons, as a constituent of all atoms.

2.2 Discovery of Neutron

In 1932 Chadwick proposed the existence of Neutron as a result of his studies in alpha particle collisions.

2.3 Black Body Radiation

1859 Gustav Kirchhoff’s studies in blackbody radiation showed that the energy radiated by a black body depended on the temperature of the body. Attempts to explain this shape of the energy and the wavelength at which the maximum energy occur continued for several decades

In 1879 Josef Stefan found that the total energy radiated by the black body per unit area is given by:

In 1884, Ludwig Boltzmann derived Stefan’s Law theoretically

In 1896, Wilhelm Carl Werner Otto Fritz Franz Wien (1864-1928) Prussia-Germany derived a distribution law of radiation.

In 1900 Max Karl Ernst Ludwig Planck (Germany 1858-1947), who was a colleague based his quantum hypothesis to explain the fact that Wien's law, while valid at high frequencies, broke down completely at low frequencies.

“Moreover, it is necessary to interpret U_N [the total energy of a blackbody radiator] not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts”
Planck. On the Law of Distribution of Energy in the Normal Spectrum. Max Planck. Annalen der Physik 4 (1901): 553.

In 1900, Planck devised a theory of blackbody radiation, which gave good agreement for all wavelengths. In this theory the molecules of a body cannot have arbitrary energies but instead are quantized - the energies can only have discrete values. The magnitude of these energies is given by the formula

E = nhf

where n = 0,1,2,... is an integer, f is the frequency of vibration of the molecule, and h is a constant, now called Planck's constant:

h = 6.63 x 10^{- 34} J $\displaystyle\cdot$ s

Furthermore, he postulated that when a molecule went from a higher energy state to a lower one it emitted a quanta (packet) of radiation, or photon, which carried away the excess energy.

With this photon picture, Planck was able to successfully explain the blackbody radiation curves, both at long and at short wavelengths. Using statistical mechanics, Planck derived an equation similar to the Rayleigh-Jeans equation, but with the adjustable parameter h. Planck found that h = 6.63 x 10^-34 J·s, fitted the data. As we can see, h is a very very small number. Thus the electromagnetic waves (light) consists not of a continuous wave but discrete tine packets of energy E = hf where f is the frequency of the light.

2.4 Photoelectric Effect

1905 when Einstein extended the photon picture to explain, another phenomenon called photoelectric effect. In this effect when light is allowed to fall on a metal and electrons are released. However there is a lower cut off frequency below which every electron stopped. Einstein was able to explain this assuming that photons are particles of energy E=hf.

Photoelectric Effect Graph - Stopping Potential vs. Light Frequency for different metals

2.5 Hydrogen Spectrum

1913 Niels Bohr (1885-1962) was able to explain the discrete spectrum of hydrogen atom with the assumption that there are possible stable energy levels where electrons can stay without emitting any wave and the light is emitted when it falls from a higher level to a lower level. The frequency of the light so emitted was given by Energy of the difference in levels = hf.

2.6 Compton Effect

In 1923, Arthur Compton showed that he could explain the collision of a photon with electrons at rest using the same idea. These phenomena came to be known as Compton Effect

compton.jpg (8608 bytes)

2.7 Wave-Particle Duality

Thus, it appears that light could behave like a wave some time (to explain reflection, refraction and polarization, interference) while at other times (Photoelectric effect, Compton effect) it behaved like a particle. The wave-particle duality of electromagnetic wave is a fact of experience and seemed mutually exclusive without compromise.

In 1924 in his doctoral thesis, Prince Louis de Broglie argued that if light waves exhibited the particle properties, particles might exhibit wave properties. The experiment to test was done on a stream of electrons as particles at a double slit and single slit and the pattern exhibited fitted the interference pattern for a wave given by

$\displaystyle{\frac{h}{mv}}$

m= mass v = speed of the electron thus: mv = momentum of the electron

2.8 Schrodinger Equation

In 1926, Erwin Schrödinger introduced operators associated with each dynamical variable and the Schrodinger equation, which formed the foundation of modern Quantum Theory. A partial differential equation describes how the wave function of a physical system evolves over time. In the Schrodinger picture differential calculus was used.
The time-independent one-dimensional Schrödinger equation is given by

$\begin{displaymath}-\frac{\hbar^2}{2m} \nabla^2\psi({\bf r}) + V({\bf r}) \psi({\bf r}) = E \psi({\bf r}) \end{displaymath}$

The solution for the value of E gives us a spectrum of values for the Energy of the system.

Using the spherical coordinates, this equation gives:

And using the separable form of the wavefunction in terms of the radial, angular parts in three dimensions

And using the potential energy as:

It gave the correct energy levels and correct spectral frequencies because of transitions. This indeed was the greatest success of Quantum Theory and which gave it the impetus.

2.9 Operators and Quantum Mechanics

In quantum mechanics, physical observables (e.g., energy, momentum, position, etc.) are represented mathematically by operators. For instance, the operator corresponding to energy is the Hamiltonian operator

$\begin{displaymath}\hat{H} = - \frac{\hbar^2}{2} \sum_i \frac{1}{m_i} \nabla_i^2 + V \end{displaymath}$

Where i is an index over all the particles of the system.

Later Dirac developed the Matrix method and is known as Dirac Bracket formalism.

In this mechanism the operators are replaced by matrices and the wave equation then reduce to a matrix equation

2.10 Quantum Wave Functions and State Vectors

While operators represent the observables, the operand – the function on which the operators act is known as the wavefunction , which is a function of the position for stationary solutions.

Postulates of Quantum Mechanics were developed later as below:

Postulate 1. The state of a quantum mechanical system is completely specified by a function that depends on the coordinates of the particle(s) and on time. This function, called the wave function or state function, has the important property that is the probability that the particle lies in the volume element located at at time t.

Postulate 2. All observables are associated with a hermitian operator. In any measurement of the observable associated with operator , the only values that will ever be observed are the eigenvalues a, which satisfy the eigenvalue equation

The solution to the eigenvalue problem given above will give a spectrum of possible values for a corresponding to a spectrum of eigenfunctions . These eigenfunctions form a set of linearly independent functions. At any point in time, we could assume that the state of the system will be a linear combination of these functions.

Some commonly used operators are given below:

Postulate 3. If a system is in a state described by a normalized wave function , then the average value of the observable corresponding to is given by

Postulate 4. The time evolution of system is given by

$\begin{displaymath}\Psi({\bf r}, t) = e^{- i {\hat H} t / \hbar} \Psi({\bf r}, 0). \end{displaymath}$

The postulates of quantum mechanics, written in the bra-ket notation, are as follows:

1. The state of a quantum-mechanical system is represented by a unit ket vector | ψ>, called a state vector, in a complex separable Hilbert space.

2. An observable is represented by a Hermitian linear operator in that space.

3. When a system is in a state |ψ1>, a measurement of an observable A produces an eigenvalue a :
A| ψ1> = a | ψ1> so that < ψ|A| ψ1> = a < ψ| ψ1> = a since the wavefunctions are orthogonal
The probability of getting this value in any measurement is

|< ψ |ψ1>|²

where | ψ1 > is the eigenvector with eigenvalue a. After the measurement is conducted, the state is | ψ1 >.

4. There is a distinguished observable H, known as the Hamiltonian, corresponding to the energyof the system. The time evolution of the state vector |ψ(t)> is given by the Schrödinger equation:

i (h/2π) d/dt |ψ(t)> = H |ψ(t)>

2.11 Heisenberg’s Uncertainty Principle

1927 Heisenberg discovered that there is an inherent uncertainty if we try to measure two conjugate observables. This is known as Heisenberg’s Uncertainty Principle

The simultaneous measurement of two conjugate variables (such as the momentum and position or the energy and time for a moving particle) entails a limitation on the precision (standard deviation) of each measurement. Namely: the more precise the measurement of position, the more imprecise the measurement of momentum, and vice versa. In the extreme case, absolute precision of one variable would entail absolute imprecision regarding the other.

” The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.”

“I believe that the existence of the classical "path" can be pregnantly formulated as follows: The "path" comes into existence only when we observe it.”

“In the sharp formulation of the law of causality-- "if we know the present exactly, we can calculate the future"-it is not the conclusion that is wrong but the premise. “

--Heisenberg, in uncertainty principle paper, 1927
http://www.aip.org/history/heisenberg/

In 1929, Robertson proved that for all observables (self-adjoint operators) A and B

Where [A,B] = AB – BA

In 1928, Dirac introduced his Bracket notation and QT in terms of matrix algebra

In 1932, von Neumann put quantum theory on a firm theoretical basis on operator algebra.

2.12 Quantum Non-locality

In 1935 Einstein, with his collaborators Boris Podolsky and Nathan Rosen, published a list of objections to quantum mechanics, which has come to be known as "the EPR paper" One of this was the problem of nonlocality. The EPR paper argued that "no real change" could take place in one system because of a measurement performed on a distant second system, as quantum mechanics requires because it will violate the relativity principles.
Einstein, B. Podolsky, N. Rosen: "Can quantum-mechanical description of physical reality be considered complete?" [i]
For example, consider a neutral-pi meson decaying into electron – positron pair. The spin of Pi meson is zero. Therefore, the total spin of electrons must be zero. Hence one of the electron will have spin (1/2) and the other spin (- ½). If the electron pair moves apart a million light years and we measure the spin of the electron on earth as ½, QM requires that the other should have a spin (-1/2) if someone measures it in his or her galaxy at the same time. How would they know which spin should it be since Relativistically it is impossible to transfer any information with a speed greater than that of light. This is the “spooky action-at-a-distance” paradox of QM

There are two choices.
You can accept the postulates of QM as is without trying to explain it, or you can postulate that QM is not complete, that there was more information available for the description of the two-particle system at the time it was created, and that you just didn't know it because QM does not properly account for it.

So, EPR requires that there are hidden variables in the system, which if known could have accounted for the behavior. QM theory is therefore incomplete, i.e. it does not completely describe the physical reality. In 1952, David Bohm introduced the notion of a "local hidden variable" theory, which tried to explain the indeterminacy in terms of the limitation of our knowledge of the complete system. [ii]

In 1964, John S. Bell, a theoretical physicist working at the CERN laboratory in Geneva proposed certain experimental tests that could distinguish the predictions of quantum mechanics from those of any local hidden-variable theory These involved the use of entangled photons – photons which interacted together at some point before being separated. These photon pair can be represented by one wave function. In 1982, Aspect, Grangier and Roger at the University of Paris experimentally confirmed that the “preposterous” effect of the EPR Paradox, the "spooky action-at-a-distance" is a physical reality. All subsequent experiments established the existence of non-locality as predicted by Quantum Theory.. [iii]

In 1986, John G Cramer of University of Washington presented his Transactional Interpretation for Quantum Mechanics.[iv]

In 1991, Greenberger–Horne–Zeilinger (GHZ) sharpened Bell's result by considering systems of three or more particles and deriving an outright contradiction among EPR's assumptions. They showed a situation involving three particles where after measuring two of the three, the third becomes an actual test contrasting between locality and the quantum picture: a local theory predicts one value is inevitable for the third particle, while quantum mechanics absolutely predicts a different value. Bell-GHZ showed that wave functions "collapse at a distance" as surely as they do locally.[v]