De Broglie equation

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source :http://switkes.chemistry.ucsc.edu

According to De Broglie, a moving small particle like electron, proton, neutron, dust particle, a small ball etc. has the properties of a wave (Dual nature of matter).

The wave length of a moving particle may be calculated as

**λ**=h/mv =h/p

De Broglie equation

**λ** =wave length

h= planck’s constt.

m=mass of particle

v=velocity of particle

p=momentum of the particle

**Dual nature of electrons –** Moving electrons in an orbit have the properties of both particle & wave. The following facts confirm the dual nature of electron.

- Cathode rays (beam of electrons) rotate a pin whell placed in their path. This property shows the particle nature of electron.
- A beam of electrons (Cathode rays) show diffraction & interference phenomenon (wave nature). This confirms the wave nature of moving electrons.

**Dual nature of Radiations**

Radiations consist of photons (Packets of energy). These photons possess the property of the wave & particle energy of photon,

**According to Planck,**

E= h*v* (1)

**According to Einstein’s equation,**

E=mc^{2 } (2)

m= mass of photon

c= Velocity of radiation

From eq. (1) & (2)

h*v* =mc^{2}

*v* =c/ **λ**

hc/ **λ** =mc^{2}

**λ** =h/mc

**Solved Numerical Problems**

**Question 1. An electron of mass 9.1×10 ^{–28} gm. is moving with velocity of 3×10^{10} cm/sec. Find its wave length?**

**Ans. ** m=9.1×10^{–28} gm. h= 6.6×10^{–27} erg-Sec

v= 3×10^{10} cm/Sec

**λ** =h/mc

=6.6×10^{–27}/9.1×10^{–28} ×3×10^{10} = 0.2417 ×10^{–9}

**λ** =2.42×10^{–10}cm Ans.

**Question 2. Mass of Electron is 9.1×10 ^{–31}kg. If its kinetic energy is 3×10^{–25} Joule, then calculate its wave length?**

**Ans. ** De Broglie Equation ,

**λ** =h/mc

m=9.1×10^{–31} kg

E= 3×10^{–25} Joule

K.E. or E =mv^{2}/2

putting the values of m & E,

v = *√ 3×10 ^{–25} *x 2 / 9.1×10

^{–31}

v = 0.812 x 10^{3}

h= 6.6×10^{–34} Joule-Sec

**λ** =h/mc

**λ** =6.6×10^{–34}/ (9.1×10^{–31} x 0.812 x 10^{3 )}

= 0.893×10^{–6}

**wave length = 8.93×10 ^{–7} m**