Uncertainty Principle Lecture 4 Uncertainty Principle References : Concept of Modern Physics by Arthur Beiser 2. Modern Physics by Kenneth Krane

A localized wave or wave packet: A moving particle in quantum theory Spread in position Spread in momentum Superposition of waves of different wavelengths to make a packet Narrower the packet , more the spread in momentum Basis of Uncertainty Principle

Heisenberg's Uncertainty Principle ___________________________________ The Uncertainty Principle is an important consequence of the wave-particle duality of matter and radiation and is inherent to the quantum description of nature Simply stated, it is impossible to know both the exact position and the exact momentum of an object simultaneously A fact of Nature!

Heisenberg's Uncertainty Principle __________________________________ Uncertainty in Position : Uncertainty in Momentum:

Heisenberg's Uncertainty Principle - applies to all “conjugate variables” ___________________________________ Position & momentum Energy & time

Uncertainty Principle and the Wave Packet ___________________________________

Some consequences of the Uncertainty Principle ___________________________________ The path of a particle (trajectory) is not well-defined in quantum mechanics Electrons cannot exist inside a nucleus Atomic oscillators possess a certain amount of energy known as the zero-point energy, even at absolute zero.

Why is n’t the uncertainty principle apparent to us in our ordinary experience…? Planck’s constant, again!! ___________________________________ Planck’s constant is so small that the uncertainties implied by the principle are also too small to be observed. They are only significant in the domain of microscopic systems

Heisenberg Uncertainty Principle The uncertainty principle states that the position and momentum cannot both be measured, exactly, at the same time. x p  h The more accurately you know the position (i.e., the smaller x is) , the less accurately you know the momentum (i.e., the larger p is); and vice versa For Numerical Applications Historic importance Where h (6.6 x 10-34) is called Planck’s constant. As ‘h’ is so small, these uncertainties are not observable in normal everyday situations

p is more p is less Increasing levels of wavepacket localization, meaning the particle has a more localized position. In the limit ħ → 0, the particle's position and momentum become known exactly. This is equivalent to the classical particle.

Heisenberg Uncertainty Principle The wave nature to particle means a particle is a wave packet, the composite of many waves Many waves = many momentums, observation makes one momentum out of many. Principle of complementarity: The moving electron will behave as a particle or as a wave, but we can not observe both aspects of its behavior simultaneously. It states that complete description of a physical entity such as a photon or an electron can not be made in terms of only particle properties or only wave properties, but that both aspects of its behavior must be considered. Exact knowledge of complementarities pairs (position, energy, time) is impossible.

Example A pitcher throws a 0.1-kg baseball at 40 m/s So momentum is 0.1 x 40 = 4 kg m/s Suppose the momentum is measured to an accuracy of 1 % , i.e., p = 0.01 p = 4 x 10-2 kg m/s The uncertainty in position is then No wonder one does not observe the effects of the uncertainty principle in everyday life! Same situation, but baseball replaced by an electron which has mass 9.11 x 10-31 kg So momentum= 3.6 x 10-29 kg m/s and p = 3.6 x 10-31 kg m/s The uncertainty in position is then

Example: A free 10eV electron moves in the x-direction with a speed of 1.88106 m/s. assume that you can measure this speed to precision of 1%. With what precision can you simultaneously measure its position? the momentum of e- is px = m vx = 9.1110-31 kg 1.88106 m/s = 1.71 10-24 kg m/s The uncertainty  px in momentum is 1% x  h/4  px =3.1 n m Example: The speed of an electron is measured to be 5.00X10E3 m/s to an accuracy of .003%. Find the uncertainty in determining the position of this electron. Calculations yields x  3.91 10-4 m.

Applications of uncertainty principle Non-existence of electrons in the Nucleus Assume that the electron is present in the nucleus. The radius of the nucleus of any atom is of the order 5 fermi (1 fermi = 10-15 m). For the existence of electron in the nucleus, the uncertainty x in its position would be at least equal to the radius of the nucleus, i.e. uncertainty in the position According to the uncertainty principle. p  h/4x= 1.05410-20 kg-m/sec If this is the uncertainty in momentum of the electron then the momentum of the electron must be at least of the order of its magnitude, that is , p  1.05410-20 kg-m/sec, an electron having so much momentum should have a velocity comparable to the velocity of light. Hence, its energy should be calculated by the relativistic formula E2=p2c2+mo2c4 The uncertainty principle based on the law of probability in quantum mechanics explains many facts which could not be explained by classical physics

Applications of uncertainty principle pc= 1.05410-20 3108 = 20 MeV The rest energy of electron 0.51 MeV, is very small as compared to pc. Hence second term in relativistic equation can be neglected. Thus, if the electron is the constituents of the nucleus, it should have an energy of the order of 20 MeV. But the experiment shows that no electron in the atom possesses kinetic energy more than 4 MeV. Therefore, it is confirmed that electrons do not reside inside the nucleus.

We apply the condition of minima from single slit diffraction, Experimental illustration of Uncertainty Principle: Single slit diffraction. To see more clearly into the nature of uncertainty, we consider electrons passing through a slit: We apply the condition of minima from single slit diffraction, and postulate that λ is the de Broglie wavelength. Momentum uncertainty in the y component Px=h/λ Since the electron can pass the slit through anywhere over the width ω, the uncertainty in the y position of the electron is y=ω.

which is in agreement with the uncertainty principle which is in agreement with the uncertainty principle. If we try to improve the accuracy of the position by decreasing the width of the slit, the diffraction pattern will be widened. This means that the uncertainty in momentum will increase. The uncertainty principle is applicable to all material particles, from electrons to large bodies occurring in mechanics. In case of large bodies, however , the uncertainties are negligibly small compared to the ordinary experimental errors.