Heisenberg Uncertainty Principle and Its Applications

The Heisenberg Uncertainty Principle states that you can never simultaneously know the exact position and the exact speed of an object. Why not? Because everything in the universe behaves like both a particle and a wave at the same time. 

Applications-

1. The non-existence of free electron in the nucleus.

The diameter of nucleus of any atom is of the order of 10-14m. If any electron is confined within the nucleus then the uncertainty in its position (Δx) must not be greater than 10-14m.
According to Heisenberg’s uncertainty principle, equation (1.27)

Δx Δp > h / 2π

The uncertainty in momentum is

Δp > h / 2πΔx ,  where Δx = 10-14m

Δp > (6.63X10-34) / (2X3.14X10-14)

i.e. Δp > 1.055X10-20 kg.m /s    ————–(1.30)

This is the uncertainty in the momentum of electron and then the momentum of the electron must be in the same order of magnitude. The energy of the electron can be found in two ways one is by non relativistic method and the other is by relativistic method.

Non-Relativistic method:

The kinetic energy of the electron is given by,

E = p2/ 2m

p is the momentum of the electron = 1.055X10-20 kg.m /s

m is the mass of the electron   = 9.11X10-31kg

E = (1.055X10-20)2/ (2X9.11X10-31) J

= 0.0610X10-9J

= 3.8X108eV

The above value for the kinetic energy indicates that an electron with a momentum of 1.055X10-20 kg.m /s and mass of 9.11X10-31kg to exist with in the nucleus, it must have energy equal to or greater than this value. But the experimental results on β decay show that the maximum kinetic an electron can have when it is confined with in the nucleus is of the order of 3 – 4 Mev. Therefore the free electrons cannot exist within the nucleus

Relativistic method:

According to the theory of relativity the energy of a particle is given by

E = mc2 =    (m0c2)/(1-v2/c2)1/2              ——-  (1.31)

Where m0 is the particle’s rest mass and m is the mass of the particle with velocity v.

Squaring the above equation we get,

E2 = (m02c4)/ (1-v2/c2)=  (m02c6)/ (c2-v2)           ——-  (1.32)

Momentum of the particle is given by p = mv = (m0v)/ (1-v2/c2)1/2

And p2 =(m02v2)/ (1-v2/c2)

= (m02v2c2)/ (c2-v2)

then p2c2 = (m02v2c4)/ (c2-v2)      ————(1.33)

Subtract eqation (1.33) from (1.32)

E2 – p2c2 = {(m02c4) (c2-v2)}/ (c2-v2)

= (m02c4)

or E2 =  p2c2 + m02c4                                  (1.34)

= c2(p2 + m02c2)

Substituting the value of momentum from eq(1.30) and the rest mass as = 9.11X10-31kg we get the kinetic energy of the electron as

E2 >  (3X108)2 (0.25 X 10-40 + 7.469 X 10-44 )

The second term in the above equation being very small and may be neglected then we get

E >  1.5X10-12 J

Or E >  9.4 MeV

The above value for the kinetic energy indicates that an electron with a momentum of 1.055X10-20 kg.m /s and mass of 9.11X10-31kg to exist with in the nucleus it must have energy equal to or greater than this value. But the experimental results on β decay show that the maximum kinetic an electron can have when it is confined with in the nucleus is of the order of 3 – 4 Mev. Therefore the electrons cannot exist within the nucleus.

2. Width of spectral lines (Natural Broadening)

Whenever the energy interacts with the matter the atoms get excited and the excited atom gives up its excess energy by emitting a photon of certain frequency which leads to the spectrum. Whatever may be the resolving power of the spectrometer is used to record these spectral lines no spectral line is observed perfectly sharp. The broadening in the spectral lines is observed due to the indeterminateness in the atomic energies. To measure the energies accurately the time (Δt) required is more that is Δt tends to infinity or otherwise. According to Heisenberg’s uncertainty relation

ΔE  = h / 2πΔt

Where ΔE is the uncertainty in the measurement of energies and Δt is the mean life time of the level is finite and of the order of 10-8s which is a finite value. Therefore ΔE must have a finite energy spread that means the energy levels are not sharp and hence the broadening of the spectral lines. Thus broadening of spectral line which cannot be reduced further for any reason is known as natural broadening.

Courtesy : http://elearning .vtu.ac.in

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