## Matter waves – wave-particle duality/ Dual nature of matter waves

We know that a particle can occupy a certain amount of space and contain some

quantity of mass. If its position is changing continuously then it should gain some velocity

(v), momentum (p), acceleration (a) and energy (E). So a particle can be characterized by its

- (i) mass
- (ii) velocity
- (iii) momentum
- (iv) energy.

Similarly, a wave can be characterized by its (i) amplitude (ii) frequency

(iii) wavelength (iv) period (v) phase (vi) wave velocity.

From the above analysis, it can be concluded that both particle and wave should not

contain similar characteristics. But from successful experiments like the photoelectric effect,

Crompton effect, photons are moving like particles. And from other experiments like x-ray

diffraction, these x-rays are moving like waves. So from this analysis, it can be concluded

that both particles and waves should have contained dual nature. It is known as the dual nature of

matter-wave and is proposed by de Broglie.

### de Broglie’s hypothesis or deBroglie wavelength of a particle

According to de Broglie’s dual nature of matter-wave, the matter-wave should

exhibit wave nature as well as particle nature. The waves associated with matter are known as

From Planck’s theory of radiation, consider a photon of mass ‘m’, moving with a

velocity ‘v’ and frequency ‘γ’ then the energy of a photon is written as

𝐸 = ℎ 𝛾. *(1)*

from Einstein’s mass-energy relation,

𝐸 = 𝑚 𝑐`2 *(2)*

(1) = (2), ℎ 𝛾 = 𝑚 𝑐`2. (3)

We know 𝑣 = 𝛾 𝜆

let v = c then 𝑐 = 𝛾` 𝜆

𝛾 =c/𝜆

(4) in (3), ℎ c/𝜆=mc`2

𝜆 =ℎ/𝑚𝑐

let c = v, 𝜆 =ℎ/𝑚𝑣. (5)

momentum = p = m v (6)

(6) in (5), 𝜆 =ℎ/𝑝. (7)

Due to the motion of the particle, it gains kinetic energy (E)

Kinetic energy = E = 1/2 𝑚 𝑣`2

𝐸 =𝑚`2𝑣`2/2𝑚=(𝑚𝑣)`2/2𝑚=𝑝`2/2𝑚

𝑝`2 = 2𝑚𝐸

𝑝 = √2𝑚𝐸 (8)

(8) in (7) 𝜆 =ℎ/√2𝑚𝐸 (9)

where h = Planck’s constant

m = mass of particle/photons

E = kinetic energy of the particle