Introduction to Quantum Mechanics

Quantum Mechanics arose as a result of problems that were becoming evident with classical physics in the late 1800’s and early 1900’s. These problems are as follows:

1. Blackbody Radiation. Blackbody radiation is the radiation that is given off by solids in equilibrium with their surroundings. This intensity of the radiation is a function of temperature and wavelength. Classical physics attempted to solve this problem but resulted in “ultraviolet catastrophe” because the leading classical theory describing blackbody radiation predicted an infinite amount of energy being emitted at short wavelengths. This calculation is based on classical physics in which the vibrating atoms were modeled as harmonic oscillators with a spectrum of normal modes frequencies and a continuum of allowed energies. It became clear that classical physics was not accurately representing nature.

In 1901, Max Plank found a solution to this problem by postulating that the energies of the harmonic oscillators were quantized or restricted to values:

n=0,1,2...

where =1.055 x 10^-34Joules which is now called Plank’s constant.

In 1905, Albert Einstein went one giant step further to explain the photoelectric effect in which electrons were emitted from the surface of metals when light was incident on the surface. He postulated that light itself was composed of particle-like quanta called “photons” with an energy:

In the mid-1920’s de Broglie argued that if light had particle properties, then particles had light or wave-like properties and speculated that particles had a certain wavelength:

This set the stage for Schrodinger in 1926 to propose his famous wave equation.

Bohr Atom

In 1913, Niels Bohr proposed a model for the atom that cleared up several unresolved problems such as the existence of descrete spectral lines and the fact that electrons in circular orbits as proposed by Rutherford didn’t decay into the nucleus. Bohr proposed that the angular momentum of the electrons is quantized:

n=1,2,3

From this equation, it is simple to derive that the energy levels are quantized with energies:

Bohr explained that the electrons could only be in these allowed states and not in any intermediate states. Furthermore, the descrete spectral lines were caused by electrons making a transition from one of these acceptable states to another.

Schrodinger’s Equation

In 1926 Schrodinger proposed his famous wave equation and described the hydrogen atom. There are several postulates of quantum wave mechanics:

1. Quantum Mechanics is probabilistic in nature in contrast to classical physic’s deterministic nature. This means that the actual value of some particular observable (i.e., measurable) quantity may be uncertain and only determined within a range of probable values.

Postulate 1: To an ensemble of identically prepared physical systems, one can associate a wave function or state function which contains all the information that can be known about the ensemble. This function is in general complex and it may be multiplied by an arbitrary complex number without altering its physical significance. The information that can be known about the system is called “observables” or “measurables”, such as position, momentum, spin... and is described by this function. Furthermore, obeys the Schrodinger Equation:

where U(x,y,z,t) is the potential energy and and must be finite, continuous, and single-valued for all values of x,y,z and t.

2. Postulate 2: The quantity can be interpreted as a position probability density such that the probability of finding the particle in a range dr centered around r_o is:

3. Postulate 3: The superposition principle holds true such that the dynamic states of an ensemble are linearly superposable. For example, if there are two possible states for an ensemble described by and , then any linear combination of these states is also a possible quantum state of the ensemble:

where c₁and c₂ are complex constants.

4. Postulate 4: With every dynamical (i.e., observable or measurable) variable is associated a linear operator. For example, a linear operator A associated with a dynamical variable A has the following properties:

Three important operators are the position, momentum and energy operators and can be expressed as:

5. A function is said to be an eigenfunction of an operator A if the result of A operating on is multiplied by some constant a_n called an eigenvalue:

Postulate 5: The only result of a precise measurement of the dynamical variable A is one of the real (i.e., noncomplex) eigenvalues a_n of the linear operator A.

Eventhough all of the postulates are ground breaking and have bizarre implications, postulate 5 takes the cake in my opinion. This is because this postulate hints at the important action of “measurement” and its result on the quantum state. After postulate 6, the implications of postulate 5 will be clearer.

6. Postulate 6: A wave function representing any dynamical state can be expressed as a linear combination of the eigenfunctions of A, where A is the operator associated with a dynamical variable A such that:

whereare the normalized eigenfunctions of the operator A with eigenvalues a_n. Postulate 6 in combination with postulate 5 states that if a system is initially prepared such that:

then the act of precisely measuring the dynamical variable A produces either a₁ or a₂. Subsequent measurements of A will produce the exact same result as the first measurement. Hence, the process of measurment has “collapsed” the wavefunction to either or .

7. Postulate 7: If a series of measurements is made of the dynamical variable A on an ensemble of systems, described by the wavefunction , the expectation or average value of this dynamical variable is:

The coefficients c_n in postulate 6 are called probability amplitudes such that the

action of measurement of a dynamical variable A produces a value a_n with a probability: .

Time Independent Schrodinger Equation

If the potential energy U(r,t)=U(r) (i.e., independent of time), then can be written as:

such that the energy operator E operating on produces a constant value E multiplied by .

Hence E is the precise and time independent energy of the system. The Schrodinger Equation can now be reduced to eliminate any time dependence resulting in the Time Independent Schrondinger Equation:

Examples

1. Free Particle