Photon and its properties

In this article, we will discuss about the photon and its properties:

A photon is basically a basic particle which carries with itself electromagnetic energy. The light coming from sun has different wavelengths or energy and on the basis of it, we have different regions like visible, infrared, ultraviolet and many more. But one thing is common in all these regions is photon but of different frequency so different energy.

Thus photon is basically a quantum of light.

Properties: Continue reading “Photon and its properties”

Application of Schrodinger wave equation: Particle in a box

Consider one dimensional closed box of width L. A particle of mass ‘m’ is moving in a one-dimensional region along X-axis specified by the limits x=0 and x=L as shown in fig. The potential energy of particle inside the box is zero and infinity elsewhere.

I.e Potential energy V(x) is of the form

V(x) = {o; if o<x<L

∞: elsewhere

The one-dimensional time independent Schrodinger wave equation is given by

d2ψ/dx2+ 2m/Ћ2[E-V] ψ=0                                             (1)

Here we have changed partial derivatives in to exact because equation now contains only one variable i.e x-Co-ordinate. Inside the box V(x) =0

Therefore   the Schrodinger equation in this region becomes

d2/ψ/dx2+ 2m/Ћ2Eψ=0

Or                 d2ψ/dx2+ K2ψ=0                                          (2)

Where                       k=    2mE/Ћ2 (3)

K is called the Propagation constant of the wave associated with particle and it has dimensions reciprocal of length.

The general solution of eq (2) is Continue reading “Application of Schrodinger wave equation: Particle in a box”

Wave function and its physical significance

WAVE FUNCTION

If there is a wave associated with a particle, then there must be a function to represent it. This function is called wave function.

Wave function is defined as that quantity whose variations make up matter waves. It is represented by Greek symbol ψ(psi), ψ consists of real and imaginary parts.

Ψ=A+iB

PHYSICAL SIGNIFICANCE OF WAVE FUNCTIONS (BORN’S INTERPRETATION): Continue reading “Wave function and its physical significance”

Time Independent Schrodinger Wave Equation

As discussed in the article of time dependent Schrodinger wave equation:

V=A exp[-i/Ћ(Et-px]

= A exp(-i/Ћ Et) exp(i/Ћ)

Ψ=ψ exp(-iEt/Ћ)                                                             (1)

Where Ћ = h/2π

So, ψ is a product of a time dependent function exp(-i/Ћ Et) and a position dependent function Continue reading “Time Independent Schrodinger Wave Equation”

TIME DEPENDENT SCHRODINGER WAVE EQUATION

In quantum mechanics, the wave function ψ corresponds to the variable y of wave motion. We know that the wave function for a particle is given by

Ψ(x,t)=A exp[-i(ωt-kx)]

Put ω=2πv and K=2π/h

Ψ(x,t)=A exp[ -i(2πvt-2π/h x)]                                      (1)

If E= total energy of the particle

P= momentum of the particle then

E=hv=2πЋ/p

Where Ћ = h/2π

Putting in equation (1), we get

Ψ(x,t)= A exp[-i(E/Ћ t –p/ Ћ x)]

Ψ(x,t) = A exp[-i/ Ћ(Et-px)]                     (2) Continue reading “TIME DEPENDENT SCHRODINGER WAVE EQUATION”

Origin of Quantum Physics

Broadly, there are two types of mechanics called classical mechanics and quantum mechanics. Classical mechanics or physics explained successfully motion of the objects which can either be observed directly or can be made observable by instruments like microscope. But, the classical mechanics can not explain the mechanics of subatomic particles like electron.proton,neutron etc. Then there comes in picture the quantum mechanics, which explain the mechanics of these subatomic particles successfully.

Following examples will show that classical mechanics was inadequate to give explanation of observed facts:

(a) Photoelectric effect: Continue reading “Origin of Quantum Physics”

Applications of the Heisenberg Uncertainty Principle: The Radius of Bohr’s First Orbit

In one of my earlier articles, I have discussed the one the applications of the Heisenberg uncertainty principle that is non-existence of electron in the nucleus. Let us discuss today the one more application of the Heisenberg uncertainty principle that is the determination of the radius of the Bohr’s first orbit. Let us start:

If ∆x and ∆px are the uncertainties in the simultaneous measurements of position and momentum of the electron in the first orbit, then from uncertainty principle

∆x∆px = Ћ

Where Ћ = h/2∏

Or    ∆px = Ћ /∆x                                                                 (1)

As kinetic energy is given as

K = p2/2m

Then uncertainty in K.E is

∆K =∆p2x/2m

Put equation (i) in above equation

∆K= Ћ2 /2m(∆x)2 (2)

As potential energy is given by

∆V= -1/4∏ε0 Ze2/∆x                                                      (3)

The uncertainty in total energy is given by adding equations (2) and (3), that is Continue reading “Applications of the Heisenberg Uncertainty Principle: The Radius of Bohr’s First Orbit”

de Broglie concept of matter waves: dual nature of matter

MATTER WAVES : dE-BROGLIE CONCEPT

In 1924, Lewis de-Broglie proposed that matter has dual characteristic just like radiation. His concept about the dual nature of matter was based on the following observations:-

(a)    The whole universe is composed of matter and electromagnetic radiations. Since both are forms of energy so can be transformed into each other.

(b)   The matter loves symmetry.  As the radiation has dual nature, matter should also possess dual character.

According to the de Broglie concept of matter waves, the matter has dual nature. It means when the matter is moving it shows the wave properties (like interference, diffraction etc.) are associated with it and when it is in the state of rest then it shows particle properties. Thus the matter has dual nature. The waves associated with moving particles are matter waves or de-Broglie waves.

WAVELENGTH OF DE-BROGLIE WAVES Continue reading “de Broglie concept of matter waves: dual nature of matter”

Applications of Heisenberg’s Uncertainty principle: Non-existence of electrons in the nucleus

Applications of Heisenberg Uncertainty principle

The Heisenberg uncertainty principle based on quantum physics explains a number of facts which could not be explained by classical physics. One of the applications is to prove that electron can not exist inside the nucleus. It is as follows:-

Non-existence of electrons in the nucleus

In this article, we will prove that electrons cannot exist inside the nucleus.

But to prove it, let us assume that electrons exist in the nucleus. As the radius of the nucleus in approximately 10-14 m. If electron is to exist inside the nucleus, then uncertainty in the position of the electron is given by

∆x= 10-14 m

According to uncertainty principle,

∆x∆px =h/2∏

Thus                            ∆px=h/2∏∆x

Or                               ∆px =6.62 x10-34/2 x 3.14 x 10-14

∆px=1.05 x 10-20 kg m/ sec

If this is p the uncertainty in the momentum of electron ,then the momentum of electron should be at least of this order, that is p=1.05*10-20 kg m/sec.

An electron having this much high momentum must have a velocity comparable to the velocity of light. Thus, its energy should be calculated by the following relativistic formula

E=  √ m20 c4 + p2c2

E =  √(9.1*10-31)2 (3*108)4 + (1.05*10-20)2(3*108)2

= √(6707.61*10-30) +(9.92*10-24)

=(0.006707*10-24) +(9.92*10-24)

= √9.9267*10-24

E= 3.15*10-12 J

Or                                  E=3.15*10-12/1.6*10-19 eV

E= 19.6* 106 eV

Or                                  E= 19.6 MeV

Therefore, if the electron exists in the nucleus, it should have an energy of the order of 19.6 MeV. However, it is observed that beta-particles (electrons) ejected from the nucleus during b –decay have energies of approximately 3 Me V, which is quite different from the calculated value of 19.6 MeV. Second reason that electron can not exist inside the nucleus is that experimental results show that no electron or particle in the atom possess energy greater than 4 MeV.

Therefore, it is confirmed that electrons do not exist inside the nucleus.

Heisenberg uncertainty principle

Statement: According to Heisenberg uncertainty principle, it is impossible to measure the exact position and momentum of a particle simultaneously within the wave packet.

We know, group velocity of the wave packet is given by

vg =∆ω/∆k

Where ω is the angular frequency and k is the propagation constant or wave number

But vg is equal to the particle velocity v

Thus vg = v =  ∆ω/∆k                                                   (1)

But                    ω=2пf

Where f is the frequency

Therefore   ∆ ω = 2п ∆ f                          (2)

Also                       k=2 п/λ

Since         de-Broglie wavelength   λ=h/p

By putting this value in equation of k, we get

k=2пp/ λ

Therefore                ∆k=2п∆p / λ                                 (3)

Put equations (2) and (3) in equation (1), we get

v= 2пh∆f/2п∆p =h∆f /            (4)

Let the particle covers distance ∆x in time ∆t, then particle velocity is given by

v  = ∆x/∆t                     (5)

Compare equations (4) and (5), we get

∆x/∆t=h∆f/∆p

Or                          ∆x.∆p=h∆f ∆t                                     (6)

The frequency ∆f is related to ∆t by relation

∆t≥ 1/∆f                                           (7)

Hence equations (6) becomes

∆x.∆p≥ h

A more sophisticated derivation of Heisenberg’s uncertainty principle  gives

∆x.∆p=h/2п                                          (8)

Which is the expression of the Heisenberg uncertainty principle.

As the particle is moving along x-axis. Therefore, the momentum in equation (8) of Heisenberg’s uncertainty principle should be the component of the momentum in the x-direction, thus equation Heisenberg’s uncertainty principle can be written as,

∆x.∆px=h/2п                             (9)

Note: There can not be any uncertainty if momentum is along y direction.

Q: Why there is uncertainty in position and momentum?

Answer: Because the particle is always in disturbed state during motion. It is not possible to calculate the position and momentum of particle simultaneously.