Basics of Quantum Mechanics

Quantum Mechanics:

Quantum state is a conglomeration of several possible outcomes of a measurement of physical properties. Quantum Mechanics is the study of matter and its interactions with energy on the scale of atomic and subatomic particles. A particle is an indivisible mass point object that has a variety of properties that can be measured, which we call observables. The observables specify the state of the particle.

A system is a collection of particles, which interact among themselves via internal forces. It can also interact with the outside world via external forces. The state of a system is a collection of the states of the particles that comprise the system. Quantum mechanics uses the language of probability theory. The basics of quantum mechanics are given below:

1. Blackbody Radiation:

Blackbody Radiation is an example of a major contradiction between theory and experiment in classical physics. A blackbody is defined in classical physics as an object that absorbs all electromagnetic radiation that falls on it. This means that a blackbody doesn’t reflect any radiation nor does it allow any radiation to pass through it. The abilities of a blackbody to absorb radiation and to reemit it are closely related to each other.

A blackbody is capable of absorbing all wavelengths of electromagnetic radiation so it is also capable of emitting all wavelengths of electromagnetic radiation. Thus a blackbody is also an ideal source of radiation, known as Blackbody Radiation.

When a blackbody is cold, it doesn’t emit any radiation. As it heats up, it starts emitting radiation. The wavelength of the emitted radiation depends only on the temperature of the blackbody, not its composition. The energy emitted per unit area is known as the intensity of radiation.

When a blackbody is heated, the electrons within it move in random directions, thus producing electromagnetic radiation. As a blackbody heats up, it moves through the spectrum becoming red, orange, yellow, green, and then blue, regardless of its composition.

At the beginning of the 20th century, two British scientists Raleigh and Jeans tried to analyze the spectrum of blackbody radiation. They were primarily interested in determining how much of the emitted radiation is stored as blue light, how much as red light and so on. They derived a formula that describes the intensity of radiation w a function of frequency f for a fixed temperature T is –

w(f, T) ∝ f2T

∴ w(f, T) ∝ T/λ2

2. Plank’s Constant:

The development of quantum mechanics began in 1900 when Max Planck found the correct explanation for the blackbody radiation spectrum. In a paper on blackbody radiation, he proposed that radiation need not be considered a continuous wave. Instead, it could be assumed to be composed of smaller chunks which he termed “quanta“.

He assumed each such quanta has an energy E that is proportional to the frequency of radiation f with a constant of proportionality (h = 6.626075 x 10-34 J.s), which was later named Planck’s constant in his honour.

[no-highlight]∴ E = hf

3. Photoelectric Effect:

In 1887, Hertz noted that light incident on a clean metal plate in a vacuum emits electrons. The electrons ejected upon the metal’s surface absorb the energy contained in the incident light. According to Maxwell’s wave theory of light, the intensity of the incident light determines the number of electrons emitted from the metal plate.

However, the energy of the emitted electrons is independent of the intensity of the incident radiation. No emission takes place if the frequency is below a certain threshold value, which is determined by the composition of the metal plate.

A few years later, Einstein showed that light consists of a stream of quanta, which he called light quanta which are called protons. Protons are electrically neutral and don’t have a mass. However, a photon has an energy equal to E as in quanta proposed earlier by Planck, and travels at the speed of light c:

[no-highlight]E = hf

The momentum p of a photon, according to the theory of relativity is –

[no-highlight]p = E/c

Since, [no-highlight]E = hf[/no-highlight],

[no-highlight]p = hf/c

To reexpress this in terms of wavelength substitute c = λf, where λ is the wavelength of the light wave. Thus,

[no-highlight]∴ p = hf/λf = h/λ

To remove an electron from a metal surface, a minimum energy Φ is called the work function. It must be expanded. Suppose a single photon of energy hf is absorbed by an electron on a metal surface. Then:

1. If hf <Φ, the electron can't be dislodged because it does not have the necessary energy to overcome the work function. 2. If hf>Φ, the electron has the energy to escape from the metal surface, and any additional energy is used as kinetic energy by the electron. Mathematically, this can be written as:

[no-highlight]hf = Φ + kinetic energy of the emitted electron

= Φ + 1/2 mv2

∴ 1/2 mv2 = Φ – hf

where m is the rest mass of the photoelectron and v is the velocity.

The above equation shows that the kinetic energy of the photoelectron depends only on the frequency of the incident radiation, not on its intensity. Furthermore, a photoelectron is emitted as a result of the interaction of a single electron with a single photon in the incident radiation wave, not the whole wave. The photoelectric effect proved clearly that light is composed of particles. This result was unexpected because until then, light was considered to behave only as a wave.