de BROGLIE’S EQUATION
de Broglie had arrived at his hypothesis with the help of Planck’s Quantum Theory and Einstein’s
Theory of Relativity. He derived a relationship between the magnitude of the wavelength associated
with the mass ‘m’ of a moving body and its velocity. According to Planck, the photon energy ‘E’ is
given by the equation
E = hν ...(i)
where h is Planck’s constant and v the frequency of radiation. By applying Einstein’s mass-energy
relationship, the energy associated with photon of mass ‘m’ is given as
E = mc2 ...(ii)
where c is the velocity of radiation
Comparing equations (i) and (ii)
mc2 = hν = c h
λ
⎛ ⎞ c ⎜ ⎟ ν = ⎝ ⎠ λ
∵
or mc = h
λ ...(iii)
or mass × velocity = wavelength
h
or momentum (p) = wavelength
h
or momentum ∝ 1
wavelength
The equation (iii) is called de Broglie’s equation and may be put in words as : The momentum
of a particle in motion is inversely proportional to wavelength, Planck’s constant ‘h’ being the
constant of proportionality.
The wavelength of waves associated with a moving material particle (matter waves) is called de
Broglie’s wavelength. The de Broglie’s equation is true for all particles, but it is only with very small
particles, such as electrons, that the wave-like aspect is of any significance. Large particles in motion
though possess wavelength, but it is not measurable or observable. Let us, for instance consider de
Broglie’s wavelengths associated with two bodies and compare their values.
(a) For a large mass:
Let us consider a stone of mass 100 g moving with a velocity of 1000 cm/sec. The de Broglie’s
wavelength λ will be given as follows :
λ =
27 6.6256 10
100 1000
− ×
× momentum
⎛ ⎞ h ⎜ ⎟ λ = ⎝ ⎠
= 6.6256 × 10– 32 cm
This is too small to be measurable by any instrument and hence no significance.
(b) For a small mass:
Let us now consider an electron in a hydrogen atom. It has a mass = 9.1091 × 10– 28 g and moves
with a velocity 2.188 × 10– 8 cm/sec. The de Broglie’s wavelength λ is given as
λ =
27
28 8
6.6256 10
9.1091 10 2.188 10
−
− −
×
×× ×
= 3.32 × 10– 8 cm
This value is quite comparable to the wavelength of X-rays and hence detectable.
It is, therefore, reasonable to expect from the above discussion that everything in nature possesses
both the properties of particles (or discrete units) and also the properties of waves (or continuity).
The properties of large objects are best described by considering the particulate aspect while
properties of waves are utilized in describing the essential characteristics of extremely small objects
beyond the realm of our perception, such as electrons.
Subscribe by Email
Follow Updates Articles from This Blog via Email
No Comments