Phase velocity and group velocity

Phase velocity of single de-Broglie wave:

The velocity of a complete wave profile is known as phase velocity or it is also defined as the speed of constant phase on the wave.

Mathematically,

Phase velocity \((V_{ph})= \frac{Wavelength(\lambda)}{Time\;Period (T)}\)

$$V_{ph}= \frac{\lambda}{T}\dotsm(1)$$

Also, $$V_{ph}=\lambda\nu\dotsm(2)$$

Where, \(\nu= Frequency = \frac1T\)

Again,

$$V_{ph}=\frac{2\pi\nu}{(\frac{2\pi}{\lambda})}$$

$$\therefore\;\; V_{ph}= \frac{\omega}{k}\dotsm(3)$$

It is also defined as the ratio of angular frequency to wave number.

Also,

$$V_{ph}=\frac{h\nu}{(\frac{h}{\lambda})}$$

$$\therefore\;\; V_{ph}= \frac{E}{P}\dotsm(4)$$

Phase velocity can also be defined as ratio of energy to linear momentum.

Case I:Phase velocity for free particle

$$E= \frac12 mv^2 $$

Where,

m= mass of particle

v= particle velocity

\(v_{ph}\)= Wave velocity

$$p= mv$$

$$\therefore \; v_{ph}= \frac{\frac12 mv^2}{mv}=\frac v2$$

$$i.e.\;\;\; v_{ph}=\frac v2\dotsm(5)$$

\(\therefore\) Phase velocity= half of particle velocity

Case 2: For bounded particle

$$E=\frac12 mv^2+ V(v)$$

$$P= mv$$

$$\therefore\; V_{ph}= \frac{\frac12 mv^2}{mv}+ \frac{V(v)}{mv}$$

$$V_{ph}=\frac v2 + \frac{V(v)}{mv}\dotsm(6)$$

Case 3: For relativistic particle

$$E=mc^2$$

where, m=dynamic mass

$$p=mv$$

$$\therefore\;\; V_{ph}=\frac EP$$

$$=\frac{mc^2}{mv}$$

$$=\frac{c^2}{v}$$

$$=\frac cv\times c$$

$$V_{ph}= \biggl(\frac cv\biggr)\times c> C$$

The phase velocity of single de-Broglie wave comes out to be greater then speed of light in relativistic case and less then particle velocity in Non-relativistic free particle. So phase velocity of a de-Broglie wave is physically meaningless quantity. It could not represent the speed of information associated with particle [ mass, energy, linear momentum , spin etc].

Group Velocity (\(V_g\))

Wave packet:

A single de-Broglie wave couldn't represent a moving particle because it's phase velocity does not represent the velocity of information associated with particle.

Schrodinger proposed that a moving material particle is associated with a large number of waves having different wavelengths and frequencies.

The wave are so selected that they interfere constructively in a region where the probability of finding a particle is maximum and destructively outside that region. The wave group so selected is known as wave packet.

The speed of center of wave packet is known as group velocity ( \(V_g\)). The group velocity represents the velocity of all information (mass, energy, momentum ) of the particle. It is equal to particle velocity.For a localized wave packet we take infinite number of wave.

The phase velocity of modulated amplitude or amplitude of wave packet is known as group velocity (\(V_g\)).

Reference:

1. Mathews, P.M and K Venkatesan. A Text Book of Quantum Mechanics. New Delhi: Tata McGraw Hill Publishing Co. Ltd, 1997.
2. Merzbacher, E. Quantum Mechanics . New York: John Wiley, 1969.
3. Prakash, S and S Salauja. Quantum Mechanics. Kedar Nath Ram Nath Publishing Co, 2002.
4. Singh, S.P, M.K and K Singh. Quantum Mechanics. Chand & Company Ltd., 2002.