The transmission of the particle is measured as a probability of the particle transmission through the barrier and is given by

Let us take a potential step as and incident particle .

Consider the beam of particles with energy E incident from left to right. In the case of we have the Schrodinger equations in region I and II as;

Region I:

…… (1)

Region II:

…….. (2)

The solution of (1) and (2) are

Since no particle coming from the right D = 0;

The boundary condition are:

and

Applying boundary condition we get,

…….. (3)

and

or ……… (4)

Adding equation (3) and (4) we get,

Subtracting equation (3) and (4), we get,

The reflection coefficient

here

Similarly, transmission coefficient is,

When ;

When ;

For , the wave are transmitted over the step but there is non zero probability for the wave to be reflected unlike the classical result.

—- (1)

The uncertainty principle usually describe these situations

1. It is impossible to predict states in which position and momentum are simultaneously arbitrarily well localized.
2. It is impossible to measure position and momentum simultaneously.
3. It is impossible to measure momentum without disturbing position and vice versa.

Hence we can say that even though a particle may contain definite momentum and coordinate, the uncertainty principle prevents us from measuring them simultaneously with considerable accuracy. Equation (1) above implies that a particle cannot simultaneously have accurate momentum and position. This inaccuracy is a natural consequence and not about imperfection of our measuring instruments.

The uncertainty principle is the manifestation of observer effect. The measurement of the position necessarily disturbs the momentum of particle.

“Physicists of today have learn that not every question about the motion of an electron or a photon can be answered but only those questions which are compatible with the uncertainty principle.” – Max Born

Putting this value in above equation we get

—— (1)

here

De Broglie assumed that equation (1) is valid for material particles like electrons. We can generalize each material particle with momentum behaves like a group of matter waves whose wavelength and wave vector k such that

——- (2)

Equation (1) and (2) are known as de Broglie relation which connects momentum of particle with the wavelength and wave vector of the wave corresponding to the particle.

a. An electron cannot revolve around the nucleus in all possible orbits as suggested by classical mechanics but  the electron can revolve round the nucleus only in those allowed orbits for which the angular momentum of the electron is an integral multiple of where

b. An atom radiates energy only when an electron jumps from a stationary orbit of higher energy to one of lower energy. If the electron jumps from an initial orbit of energy to a final orbit of energy and then a photon with frequency is emitted.

Mathematical approach for Bohr theory of atom

Let us consider an atom with one electron of mass m moves with velocity in a circular orbit of radius centered at the nucleus. The charge of the electron is and that of the nucleus is . The centripetal force is equal to the coulomb force between electron and nucleus.

… (a)

The kinetic energy of the electron is

… (b)

According to Bohr, the stationary states of the system are characterized by definite values of the angular momentum such that,

n=1,2,3,……… ; … (c)

The velocity of the electron in its orbit is;

… (d)

Let us put equation (d) in equation (b) and the radius of the orbit is obtained as;

… (e)

For the ground state of hydrogen atom n=1 and Z=1; the radius of the orbit is,

This quantity is known as Bohr radius.

The total energy of the atom is the sum of kinetic and potential energies.

The energy of the system in its state is;

…. (f)

As the value on n increases increases and the result is that the outer orbits have greater energies than the inner orbits.

Spectral series

If an electron jumps from higher energy orbit to lower energy orbit , the frequency of energy emitted is given by

.

Here using equation (f)

and

Therefore

Wave number of a radiation is defined as the reciprocal of its wavelength in vacuum;

Here

is the Rydberg constant.

Hydrogen spectrum is produced when the electron falls from outer orbit into innermost orbit. The five spectral series of hydrogen atom, named for their discoveries are
Lyman series (ultraviolet region)
Balmer series (visible region)
Paschen series (infrared region)
Brackett series (infrared region)
P fund series (infrared region)

Einstein was the scientist behind more detail and elaborate explanation of photoelectric effect. He postulated his theory on photoelectric effect in 1905 and said that light consist of particles called photon. Each photon has an energy of and the linear momentum where is frequency of wave and is the wavelength of the light.

When a beam of light of frequency is incident on a metal, each photon transmits all its energy to an electron neat the surface. In this case the photon is entirely absorbed by the electron. Thus the electron will absorb energy only in quanta of energy and intensity of incident radiation doesn’t have any effect. There is always a minimum energy which is required to knock off electron from the metal and this energy is called work function . If is larger than the electron will be knocked off and the electron will not jump if this condition is not satisfied. For electron come out with kinetic energy with the maximum value such that

where represents the kinetic energy of the electron leaving the material. This is Einstein photoelectric equation and gives explanation to photoelectric effect and is validated by experimental observation. From this equation we can see that the ejected electron’s kinetic energy increases linearly with the incident frequency .

Here =

Here and is called threshold or cutoff frequency of the metal. This relation shows that no electrons are ejected from the metal unless and the frequency of radiation is important rather than the intensity of the radiation to knock off electron. The kinetic energy of the knocked off electron is acquired from the excess energy given by the incident radiation.

]]> https://physicsanduniverse.com/photoelectric-effect/feed/ 0 8189 https://physicsanduniverse.com/blackbody-radiation/ https://physicsanduniverse.com/blackbody-radiation/#respond Sun, 20 Sep 2015 07:25:38 +0000 http://physicsanduniverse.com/?p=8139 A blackbody is an object that is perfect absorber of radiation. It is a material object that absorbs all radiation falling on it and hence appears black under reflection. When an object is heated, it radiates electromagnetic energy as a result of thermal agitation of electrons on its surface. The intensity of this radiation depends on its frequency and temperature. This emitted light ranges over the entire spectrum. An object in thermal equilibrium with its surrounding radiates as much as it absorbs. It thus follows that a blackbody is a perfect absorber as well as a perfect emitter of radiation.

A blackbody is constructed by taking a hollow cavity whose internal walls are constructed to be perfectly reflective of electromagnetic radiation and it has a very small hole on its surface. The radiation entering from the hole is trapped inside the cavity and gets completely absorbed after many reflections on the inner cavity wall. As absorption continues, temperature is raised and the hole will eventually begin to glow. This hole will now become perfect emitter and the radiation emitted in such condition is called blackbody radiation.

It is to be noted that the peak of the radiation spectrum occurs at a frequency that is proportional to the temperature . This is the reason why a heated body emit radiation from red to yellow to white as temperature increases. This means that the peak would shift more to the right and classical theory was unable to explain the high frequency behavior of blackbody emission. To explain these results, classical theory is modified in the Rayleigh-Jeans formula. The electromagnetic energy density  in the frequency ranges  to  is defined as:

where is the number of degrees of freedom for frequency and is the average energy per degree of freedom that is average energy of the oscillators present on the walls of the cavity.

According to the equipartition theorem of classical thermodynamics all oscillators in the cavity have the same mean energy, irrespective of their frequencies. Now, Rayleigh-Jeans formula is

and this formula works well at low frequencies only.

In contrast to Rayleigh’s assumption, Plank considered that the radiation of the frequency can only exchange energy with matter in discrete units of energy or quanta of energy that is the energy if the radiation of frequency emitted by the oscillating charges must come only in integral multiples of .

Formula for Planck’s distribution law is

Planck’s law is in agreement with the experimental data.

 The beginning of twentieth century brought few theories to light and challenged the ideas of classical mechanics. These ideas includes relativistic mechanics and Quantum mechanics. Classical theory failed to explain the motion of bodies traveling at speed very close to speed of light and it also failed to explain several microscopic phenomena like atomic stability, photo electric effect and blackbody radiation.

Max Plank was the biggest contributor to the beginning of quantum physics as he introduced the concept of quantum of energy to explain blackbody radiation. He postulated that the energy exchange between surrounding and radiation happens in discrete or quantized amounts. He explained that the energy exchange of electromagnetic waves of frequency with matter occurs only in integer multiples of where h is planck’s constant. This idea lead to the solution of blackbody radiation and many other problems at that time.

Influenced by idea of Planck, Einstein postulated that this idea must be valid for light as well and he  put forward the idea that light itself is made up off discrete bits of energy called photons and each photon has an energy of . Using this idea he beautifully explained photoelectric effect which has been a problem for physicist since 1887. Einstein also received Nobel Prize in Physics for this.

Neil Bohr in 1913 put forward the model of hydrogen atom by combining the concept of Photon, Planck’s quantum concept and Rutherford’s atomic model. His theory was that atoms can be found only in discrete states of energy and interaction of the atom with radiation takes place only in discrete amounts  because of the transition of atom between its various discrete energy states.

Compton in 1923 with the help of x-ray scattering experiment conformed the corpuscular aspect of light and showed that x-ray photons behave like a particles  with momenta . In 1924, de Broglie introduced an idea which associated electron with  waves. He purposed that its not only wave that behaves like particle but the converse is also  true and particles themselves behave like a wave and this was confirmed by Davisson and Garmer in 1927. By 1925 quantum mechanics was in full swing and scientist like Heisenberg and Schrodinger were in forefront of new discoveries and theories of Quantum physics.

The matrix mechanics and wave mechanics were two independent formulation of quantum mechanics. Matrix mechanics is used to describe atomic structure and was developed by Heisenberg. Heisenberg propounded his theory on the notion that the only allowed values of energy exchange between micro systems are those that are discrete quanta. Expressing quantities like angular momentum, momentum, position and energy in terms of matrices, he obtained an eigenvalue problem that describes the energy spectrum and the state vectors of the system.

The wave mechanics was formulated by Schrodinger and is more intuitive than matrix mechanics. Max Born proposed his probabilistic interpretation of wave mechanics. He took the square moduli of the wave functions that are solutions to the Schrodinger equation and the interpreted them as probability densities.

The concept of abstract objects like bras, kets and operators was introduced by Dirac.Dirac in 1928 derived an equation which describes the motion of electron. This equation predicted the existence of positron which is antiparticle of electron. This predicted particle was discovered in 1932 four years later.

How it works?
well all electrons have magnetic fields so they’re basically like tiny bar magnets and this property is called spin. If you place them in a magnetic field they will align with magnetic field just like a compass needle lines up with the magnetic field of the earth and this is the lowest energy state so you can call it 0 state or we call it for the electron spin down. Now you can put it into one state or spin up but that takes some energy.

If we try to align this electron in other state than it naturally tends to go to, we will have to give some energy and this is the highest energy state. In principle if we were able to put it exactly against the magnetic field it would stay there. So far we are just talking about classical bit concept for electron. It has two states spin up and spin down which like the classical 1 and 0 but the interesting thing about quantum objects is that they can be in both states at once now when you measure this spin it will be either up or down but before you measure it the electron can exist in what’s called a quantum superposition and the probability of finding electron in either up or down state is determined by some coefficient. This coefficient represents the relative probability of finding the electron in one state or the other.

To realize the incredible computing power of quantum computers we need to consider an interacting quantum bits. With two electrons in discussion, there are four possible states of these two electrons. At this point you might think that its just like classical computer. With two bit information we can have four possibilities that is 00 01 10 and 11. Even though its 4 possible information, it’s just 2 bit information.

Now quantum mechanics allows us to have the superposition of these four states. So to determine the state of the qubits we need four coefficient numbers. In classical example we need only two bit information (0 and 1) to know about all four states. So in extension, if we have three qubits, it can represent eight states.

Going this way we can easily realize that N qubits can contain classical bits information. It should be noted that a quantum computer can perform more operations simultaneously; reducing the time it takes to arrive at the end result by exponentially reducing the number of computation required.

Postulate 1 (describes the state of quantum mechanical system): At a given time the state of quantum mechanical system is defined by specifying a ket belonging to state space .

Postulate 2 (description of physical quantity): Every measurable quantity A is described by an observable acting in the state space .

Postulate 3 (Measurement of physical quantity): The only possible result of a measurement of a physical quantity A is one of the Eigen values of observable .

Postulate 4 (Principle of spectral decomposition) Statement 1 (Discrete non-degenerate spectrum): When a physical quantity A is measured on a quantum mechanical system in the normalized state , the probability of obtaining the non degenerate Eigen values of the observable is here and is normalized.

Postulate 4 Statement 2 (Discrete degenerate spectrum): When a physical quantity A is measured on a quantum mechanics system in the normalized state , the probability of obtaining the degenerate Eigen value of the observable is
where and is degeneracy of .

Postulate 5 (Reduction of wave packet): If the measurement of a physical quantity A on a quantum mechanical system in the state , gives the result , the state immediately after the measurement is given by the normalized projection of onto the Eigen space associated with that is $latex \frac{P_n|\psi>}{\sqrt{<\psi|P_n|\psi>}} $

Postulate 6 (Time evolution of quantum mechanical state): The time evolution of state vector is governed by the Schrodinger equation.

here is Hamiltonian of system.