Heisenberg’s Uncertainty Principle - Civil Engineering (CE) PDF Download

Heisenberg’s uncertainty Principle
 Statement: 
“It is impossible to determine both position and momentum of a particle simultaneously and accurately. The product of uncertainities involved in the determination of position and momentum simultaneously is greater or equal to h/4π ”.

Explanation: If we try to measure both position and momentum of a particle accurately and simultaneously, there will be always some error involved in the measurement. If we try to measure position accurately, error involved in the measurement of momentum increases and vice versa. If ∆_x is error (uncertainty) involved in the measurement of position and ∆p_x is the error involved in the measurement of momentum then according to Heisenbergs Uncertainty Principle.

Significance of Heisenberg’s uncertainty Principle:
We know that according to classical mechanics, if we know the position and momentum of a particle at some instant of time, its position and momentum at any later instant of time can be determined accurately. But according to Heisenberg’s uncertainty Principle, if we try to measure position accurately, error in the measurement of momentum increases and viseversa. Thus in quantum mechanics there is no place for the word exactness, and is replaced by the word probability. Heisenberg’s uncertainty Principle implies that an event, which is impossible to occur according to classical physics, has finite probability of occurrence according to quantum mechanics.

Application of Uncertainty Principle
Non existing of electron in the nucleus We know that the diameter of the nucleus is of the order of 10⁻¹⁴m. If an electron is to exist inside nucleus, then uncertainty in its position ∆x must not exceed the size of the nucleus

Then its uncertainty in the momentum is given by

Thus the momentum of the electron must be at least equal to the above value. Then minimum energy that electron should have in order to exist inside the nucleus can be calculated using the formula

Where p=momentum of the electron, c=velocity of light and m0=rest mass of the electron. Substituting the values, we get

Explanation of β-decay So, in order to exist inside the nucleus, electron should posses energy greater or equal to 10 MeV . But experimentally determined values of energy of the electrons emitted by nucleus during the beta-decay were not greater than 4 MeV . This clearly indicates that electrons cannot exist inside the nucleus.

Time independent Schrodinger wave equation
Consider a particle of mass m, moving with the velocity v along the +ve x-axis. Then according to de-Broglie’s hypothesis, wavelength of the wave associated with the particle is given by

λ = h/mv

A wave traveling along x-axis can be represented by the equation

Ψ(x, t) = A e^{−i(ωt − κx)}

Where Ψ(x, t) is called wave function. The equation (2.2) can be written as

Ψ(x, t) = Ψ(x)A e^−iωt
where Ψ(x) = e^iκx

Differentiating equation (2.3) w.r.t x twice we get.

Differentiating equation (2.3) w.r.t t twice we get

General wave equation is given by

From equation (2.4) and (2.5) we get

The total energy of the particle is given by
E = 1/2 mv² + V

where V is the potential energy
mv² = 2(E − V )
(mv)² = 2m(E − V )

substituting for (mν) from equation (2.1) we get

substituting in equation (2.6) we get

The above equation is called one-dimensional Schrodinger’s wave equation. In three dimension the Schrodinger wave equation becomes

Interpretation of wave function and its properties
The state of a quantum mechanical system can be completely understood with the help of the wavefunction Ψ. But wave function Ψ can be real or imaginary. Therefore no meaning can be assigned to wavefunction Ψ as it is. According to Max Born’s interpretation of the wavefunction, the only quantity that has some meaning is | Ψ | ² , which is called probability density. Thus if Ψ is the wavefunction of a particle within a small region of volume dv, then | Ψ | ² dv gives the probability of finding the particle within the region dv at the given instant of time.
We know that electron is definitely found somewhere in the space

I | Ψ | ² dv = 1

The wave function Ψ, which satisfies the above condition, is called normalized wave function

Nature of Eigenvalues and Eigenfunctions (Properties of wave function)
A physical system can be completely described with the help of the wave function Ψ. In order to get wavefunction, first we have to set up a Schrodinger wave equation representing the system. Then Schrodinger wave equation has to be solved to get wavefunction Ψ as a solution. But Schrodinger wave equation, which is a second order differential equation, has multiple solutions. All solutions may not represent the physical system under consideration. Those wavefunction which represent the physical system under consideration are acceptable and are called Eigenfunction.

A wavefunction Ψ can be acceptable as wavefunction if it satisfies the following conditions.

1. Ψ should be single valued and finite everywhere.
2. Ψ and its first derivative with respect to its variables are continuous everywhere.

The solution of the Schrodinger wave equation gives the wavefunction Ψ. With the knowledge of Ψ we can determine the Energy of the given system. Since all wavefunctions are not acceptable, all the values of energies are not acceptable. Only those values of energy corresponding to the Eigenfunctions which are acceptable are called Eigenvalues.