4.Twin Paradox

Twin Paradox: There is a twin brother A & B. A takes space voyage and B remain in the earth. According to time dilation, the clock with B runs faster than clock with A.

 we know that,
                      

For example,   if B is measuring a span of 50 years, A would be measuring only 30 years.      

                   A would look 20 years younger than his twin B.

Let us study the problem from the point of view of A who is in space voyage. We know that if there is a relative motion between two observers, each one thinks that the other observer is moving. Here, A would see that his twin brother B and earth moves away from him with a velocity  during his outward journey. So, A thinks that B's clock is running slower as per relativity and B is younger than A at the end of space voyage. Thus B thinks that A is younger and A thinks that B is younger and this creates a paradox.

The paradox comes from measurement made by A. According to him, he is in the inertial frame and earth moves with velocity away from him.  According to his calculation B's clock runs slower. This is not true because his frame is not perfectly inertial because he has to turn back his spaceship at one point of time to return back, so he would experience acceleration. So, all observers would agree that only A is in motion and his clock runs slower at the end of the space voyage. Actually A looks younger than B.

It has been proved that moving clock runs slower. The decay rate of fast moving radioactive materials is slower than the radioactive sample at rest.

3. Application

Introduction:

The uncertainty principle has far reaching implications. In fact, it has been very useful in explaining many observations which cannot be explained otherwise.

Application:

An important one being the proof of the non-existence of an electron inside the nucleus. In beta decay, the electrons are emitted from the nucleus of the radioactive element. The radius of a typical atomic nucleus to be about  5.0 × 10⁻¹⁵ m. Assuming that the uncertainty in the position of the electron inside the nucleus to be of the same order, we have

If this is the uncertainty in a nuclear electron’s momentum p itself must be at least comparable in magnitude. An electron with such a momentum has a kinetic energy, KE, many times greater than its rest energy (which is mc²). The kinetic energy of such particle is given by

This means that the kinetic energy of an electron must exceed 20MeV if it is to be inside a nucleus. Experiments show that the electrons emitted by certain unstable nuclei never have more than a small fraction of this energy, from which we conclude that nuclei cannot contain electrons. The electron that an unstable nucleus may emit comes into being only at the moment the nucleus decays.

2. Physical significance of a wave function:

Introduction:

In quantum mechanics, because of the wave-particle duality, the properties of the particle can be described as a wave. Therefore, its quantum state can be represented as a wave of arbitrary shape and extending over all of space. This is called a wave function. The wave function is usually complex and is represented by Ψ. Since the wave function is complex, its direct measurement in any physical experiment is not possible. It is just mathematical function of x, t etc. Once the wave function corresponding to a system is known, the state of the system can be determined. The physical state of system is completely characterized by a wave function.

Physical significance of a wave function:

The wave function contains information about the system it represents. Even though the wave function itself is not directly an observable quantity, the square of the absolute value of the wave function gives the probability of finding the particle at a given space and time. This probabilistic interpretation of wave function was given by Max Born in 1926. If Ψ is the wave function associated with a particle, the |Ψ|² is the probability per unit volume that the particle will be found at the given point. The probability density is given by

where Ψ^∗ is the complex conjugate of Ψ. For a particle restricted to move only long x− axis, the probability of finding it between x₁ and x₂ is given by

Since the probability of finding a particle any where in a given voluve must be one, we have

This condition is know as normalization.

Properties of a wave function:

A wave function has the following characteristics.

1.      Ψ must be continuous and single-valued everywhere.

2.      ∂Ψ/∂x, ∂Ψ/∂y and ∂Ψ/∂z must be continuous and single-valued everywhere.

3.      Ψ must be normalizable.

1.Introduction: Heisenberg’s uncertainty Principle

Introduction: Heisenberg’s uncertainty Principle

Quantum mechanics is a fundamental branch of physics which generalizes classical mechanics to provide accurate descriptions for many previously unexplained phenomena such as black body radiation, photoelectric effect and Compton effect. The term quantum mechanics was first coined by Max Born in 1924.Within the field of engineering, quantum mechanics plays an important role. The study of quantum mechanics has lead to many new inventions that include the laser, the diode, the transistor, the electron microscope, and magnetic resonance imaging. Flash memory chips found in USB drives also use quantum ideas to erase their memory cells. The entire science of Nanotechnology is based on the quantum mechanics. Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than the regular computers. This chapter attempts to give you an elementary introduction to the topic.

Explanation:

We know from the wave-particle duality that every particle has wave-like properties. These wave properties of particles will prevent us from measuring the exact attributes of the particles. This limitation related to the measurements at microscopic level is known as the uncertainty principle. The uncertainty principle states that it is impossible to specify simultaneously the position and momentum of a particle, such as an electron, with precision. The theory further states that a more accurate determination of one quantity will result in a less precise measurement of the other, and that the product of both uncertainties is always greater than or equal to Planck’s constant divided by 4π. That is

This principle was formulated in 1927 by the German physicist Werner Heisenberg. It is also called the indeterminacy principle.

The Heisenberg’s uncertainty principle can also be expressed in terms of the uncertainties involved in the simultaneous measurements of angular displacement & angular momentum and energy & time;

Sometimes h/2π is written as h. In that case the right had side of the uncertainty relations will have h/2.

Physical significance of uncertainty principles:

The uncertainty principle is based on the assumption that a moving particle is associated with a wave packet, the extension of which in space accounts for the uncertainty in the position of the particle. The uncertainty in the momentum arises due to the indeterminacy of the wavelength because of the finite size of the wave packet. Thus, the uncertainty principle is not due to the limited accuracy of measurement but due to the inherent uncertainties in determining the quantities involved. Even though, the uncertainty principle prevents us from knowing the precise position and momentum, we can define the position where the probability of finding the particle is maximum and also the most probable momentum of the particle. That means, the uncertainty principle introduces the probabilistic interpretation of the physical quantities. This is the major difference between the classical physics and quantum mechanics.