Introduction to Quantum Physics;Heisenberg''s uncertainty principle (Video Lecture)

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Quantum Mechanics (Uncertainity principle)

Question If a freely moving electron is localized in space to within $\Delta x_0$ of $x_0$, its wave function can be described by a wave packet $\psi(x,t)=\int_\infty^{-\infty}e^{i(kx-\omega t)}f(k)dk$, where $f(k)$ is peaked around a central value $k_0$. Which of the following is most nearly the width of the peak in $k$?
A. $\Delta k = 1/x_0$
B. $\Delta k = \frac{1}{\Delta x_0}$
C. $\Delta k = \frac{\Delta x_0}{x_0^2}$
D. $\Delta k = k_0\frac{\Delta x_0}{x_0}$
E. $\Delta k = \sqrt{k_0^2+(1/x_0)^2}$
Solution:
In quantum mechanics, the momentum $(p=\hbar{k})$ and position $(x)$ wave functions are Fourier transform pairs and the relation between $p$ and $x$ representations forms the Heisenberg uncertainty relation:
$\Delta{x}\Delta{k}\geq1 \Rightarrow \Delta k \geq \frac{1}{\Delta x}$
Answer: B

The Schrödinger wave equation

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de Brogli wavelength

• Interference, diffraction and polarization establishes the wave nature of light and can be explained on the basis of wave theory
• photoelectric effect and compton effect are explained on the basis of quantum theory of light which establishes that quanta behaves like corpescules.
• Louis de Brogli proposed that idea of dual nature i.e., wave particle duality should be extanded to all micro particlesi.e., wave and corpescular nature should be associated with each particle.
• According to him material particles might have dual nature same as that of light.
• He suggested that moving particle whatever is its nature , has wave properties associated with it.
• Wavelength λ associated with any particle of momentum p is given by
  λ=h/p=h/mv
  where h is the Plank's constant.
• From Plank's theory of radiation , energy of photon is given by
  E=hν = hc/λ
  therefore λ=hc/E
  From special theory of relativity
  E=mc²
  mass of photon is m=hν/c²
  momentum of photon is
  mc=p=hν/c = h/λ
  therefore p=h/λ
  or, λ=h/p
• Thus what is true for energy packet (photon) is also true for material particle. Thus, for particle of mass m moving with velocity v , we have p=mv and de Brogli wavelength associated with it is
  λ=h/mv
  where m is the relativistic mass of the particle
• If m or v is large the de Brogli wavelength associated with a material particle would be small.
• The de Brogli wave associated with a material particle or photon of any charge associated with it can also be calculated. Thus we know that
  K=mv²/2
  therefore mv=√(2mK)
  λ=h/√(2mK)
• If a charged particle carrying a charge q is accelerated through a potential difference V volts, then kinetic energy K=qV
  Therefore be Brogli wavelength of charge particle for charge q and accelerated through a potential difference of V volts is given by
  λ=h/√(2mqV)

Uncertainity Principle

Uncertainity principle says that "If a measurement of position is made with accuracy Δx and if the measurement of momentum is made simultaneously with accuracy Δp , then the product of two errors can never be smaller than a number of order h
ΔpΔx≥(∼h)                                  (1)
Similarly if the energy of the syatem is measured to accuracy ΔE , then time to which this measurement refers must have a minimum uncertainity given by
ΔEΔt≥(∼h)                                    (2)
In generalised sence we can say that if  Δq is the error in the measurement of any co-ordinate and Δp is the error in its canonically conjugate momentum then we have,
ΔpΔq≥(∼h)                                  (3)
Consider the relation between the range of position Δx and range of wave number Δk appearing in a wave packet then
ΔxΔk≥1                                                     (4)
and this is a general property not restricted to quantum mechanics. Uncertainity principle is obtaines when the following quantum mechanical interpretation of quantities appearing in above equation are taken into account.
(1) The de-Brogli equation p=hk creates a relationship between wave number and momentum , which is not present in classical mechanics.
(2) Whenever either the momentum or the position of an electron is measured , the result is always some definite number. A classical wave packet always covers a range of positions and range of wave numbers.
Δx is a measure of minimum uncertainity or lack of complete determination of the position that can be ascribed to the electron. and Δk is the measure of minimum uncertainity or lack of complete determination of the momentum that can be ascribed to it.

Relation of spreading wave packet to uncertainity principle
Narrower the wave packet to begin with , the more rapidly it spreads. Because of the confinement of the packet within the region Δx₀ the fourier analysis contains many waves of length of order of Δx₀ , hence momenta p≅h/Δx₀
therefore
Δv≅p/m≅h/mΔx₀
Although average velocity of the packet is equal to the group velocity , there is still a strong chance that the actual velocity will fluctuate about this average by the same amount. The distance covered by the particle is not completely determined but it may vary as much as
Δx≅tΔv≅ht/mΔx₀
The spread of the wave packet may therefore be regarded as one of the manifestations of the lack of complete determination of initial velocity necesarily associated with the narrow wave packet.

Relation of stability of atom to uncertainity principle
From uncertainity principle if an electron is localized it must have on an average a high momentum and have high kinetic energy as it takes energy to localize a particle. According to uncertainity principle it takes a momentum  Δp≅h/Δx and an energy nearly equal to h²/2m(Δx)² to keep an electron localised within a region Δx. Momentum creates a pressure which tends to oppose localization of the electron. In an atom the pressure is opposed by the force attracting the electron back to the nucleus. Thus the electron will come to equilibrium when the attractive forces balances the effective pressure and, this way , the mean radius of the lowest quantum state is determined. This point of balance can be found from the condition that total energy must be minimum. Thus we have
W≅  (h²/2m(Δx)²) - (e²/Δx)
Differentiating both the sides w.r.t. Δx and making ∂W/∂(Δx) = 0 we get
Δx≅h²/me²
THis result is just the radius of first Bohr orbit although not exact but qualitative.. The limitation of the localizability of the electron is inherent in the wave-particle nature of matter. In order to have an electron in very small space , we must have very high fourier components in its wave function and therefore the possibility of very high moments. There is no way to force an electron to occpy a well defined position and still remain at rest.