QUANTUM MECHANICS.pdf

QUANTUM MECHANICS CLASSICAL PHYSICS A set of physics theories that predate the modern theories and are used to explain the world that we see by our naked eye. Classical physics could not explain black body radiation, photoelectric effect and Compton effect. BLACK BODY RADITION When radiation falls...

QUANTUM MECHANICS CLASSICAL PHYSICS A set of physics theories that predate the modern theories and are used to explain the world that we see by our naked eye. Classical physics could not explain black body radiation, photoelectric effect and Compton effect. BLACK BODY RADITION When radiation falls on a material, it absorbs, emits and transmits the radiation. The total energy of the material will be the sum of energy absorbed, emitted and transmitted. A body that absorbs all the radiation is known as a black body. Heated objects emit thermal radiation and they glow too. Glowing solid objects emit thermal radiation whose frequency ranges continuously from infrared to ultraviolet. Under equilibrium condition, a black body emits thermal radiation whose intensity depends on the wavelength of the emitted radiation and temperature of black body. BBR AS A FUNCTION OF WAVELENGTH - CURVE 1 The peak of the graph is in visible region. As the wavelength is increased or the frequency is decreased then the peak or the spectral irradiance of the bb decreases. Spectral irradiance is also known as energy density it is the energy emitted by a bb per unit area per unit wavelength. CLASSICAL LAWS THAT WERE USED TO EXPLAIN BBR 1. Wien’s displacement law  The BBR curve for different temperatures, peaks at different wavelengths that are inversely proportional to temperature.  It helps describe how the peak of the black body radiation curve shifts with temperature but doesn't explain the entire shape of the curve. Λmax ∝ T-1 2. Stefan’s law  It states that total energy emitted per unit surface area per unit time is proportional to the fourth power of temperature.  It relates to the overall intensity of radiation emitted by the black body but not the specific shape of the radiation curve. E = σT4 Attempts were made to explain the nature of BBR curve by classical theories that were- i. Wein’s distribution law: According to Wein’s distribution law the energy density is given by – Wein’s displacement law was convenient for shorter wavelengths but failed to explain experimental data of higher wavelengths. ii. Rayleigh-Jeans law: According to Rayleigh and jean the radiation inside the cavity forms standing waves and the spectral density is given as- 2  The Rayleigh’s law agreed with experimental result for larger wavelengths but strongly disagreed for shorter wavelengths. The inconsistency between observations and predictions of classical physics is known as ultraviolet catastrophe PLANK’S LAW – quantum’s origin #Plank said that black body consists of atoms and those atoms vibrate (electric oscillators) this vibration leads to emission and absorption of electromagnetic radiation.  Electric oscillators in a bb lead to emission and absorption EM radiation.  When radiation falls on a bb, electric oscillations increase, bb contains oscillators of all frequencies and energy of these oscillators is quantized (E = nhv).  The vibrational energy of these oscillators has discrete values.  Planck assumed that radiation is a gas of massless, uncharged particles, namely photons.  Planck further assumed that when an oscillator changes from a state of higher energy to lower energy, the difference of the lower and higher energies is equal to frequency times plank’s constant (h). The average energy of a mode/oscillator of frequency ν according to quantum theory is - 3 PHOTO-ELECTRIC EFFECT  The photoelectric effect is the phenomenon in which electrons are emitted from a material (usually a metal) when it is exposed to light of a certain frequency or higher. These ejected electrons are called photoelectrons.  Photo electric effect cannot be explained by classical physics. Why? Observation – 1:  Different metals required light of different minimum frequencies for electron emission to occur.  Increasing the intensity of light (brightness of light) increased the number of electrons without increasing their energies. CLASSICAL PHYSICS – according to classical theory light is a continuous wave and its energy depends on its intensity meaning that as brightness is increased the energy of the electrons also increases, also by this theory it was assumed that a sufficiently intense light of low-frequency (frequency even lower than threshold) can emit electrons, which was definitely not true since metals require light of frequency greater than or equal to a minimum threshold. Observation – 2:  When light of sufficient frequency falls on a metal surface the electrons emit instantly. 4  CLASSICAL PHYSICS – according to classical theory the electrons need time to absorb the incoming radiation before being ejected. Observation – 3:  The emission of electrons from the metal surface or the kinetic energy of electrons solely depended upon the frequency of incoming radiation and not on its intensity.  Even a radiation of low intensity but frequency greater than the threshold ejected electrons immediately.  Intensity of radiation only affected the number of electrons. CLASSICAL PHYSICS – according to classical theory higher intensity of radiation must emit electrons of higher energies or must increase the kinetic energy of the electrons. These inconsistencies led to the development of quantum mechanics, in which light was introduced as quantized packets of energy called photons. Energy if each photon is given as ~ COMPTON EFFECT  Further explains the particle nature of light.  Arthur Compton scattered X-ray from electrons bound in atoms.  The electrons loosely bound to the atoms were considered as free electrons.  By classical theory the wavelength of X-ray would not change on interaction with electrons.  But upon observation the wavelength of X-ray was in fact changing on interaction with electron. (increasing)  This change in wavelength can be explained by treating light as a particle.  In Compton experiment X-rays are composed of high energy photons, when these photons collide with electrons of an atom then, electron absorbs some energy from the photon and is ejected out of its position while the photon loses some of its energy which leads to the change in its wavelength.  The wavelength increases after colliding with electron.  The wavelength increase is given as – 5  (Refer the derivation of recoil of an electron)  The wavelength increase depends solely on the scattering angle.  h/moc is known as the Compton wavelength, for electron its value is 2.43 x 10-3 nm  The Compton effect is observed generally for photons of high frequency.  Equation is valid for particles other than electrons if proper mass is used.  Free electrons within the metals are used in this experiment. WAVE NATURE OF PARTICLE – THE ORIGIN  The photons were known to possess wave-particle duality before de - Broglie  De Broglie extended wave-particle duality for particles such as electrons  De Broglie proposed that all matter and light, has dual nature i.e., particles can behave like both particle and wave. This meant that electrons which were traditionally thought of as particles can behave like a wave.  If particles can exhibit wave nature, then their wavelength is given as~ Λ=h/p  To verify this hypothesis Davisson Germer experiment was done DAVISSON – GERMER EXPERIMENT  Davisson and germer bombarded a nickel crystal with an electron beam. The scattered electrons exhibited a diffraction pattern which proved the wave nature of the electrons present in the beam.  Diffaraction and intereference patterns are exhibited by waves 6 COMPARISON OF WAVELENGTHS OF A PHOTON AND ELECTRON Electron has kinetic energy = 1ev Photon has energy = 1ev Photon ~ ~ Electron ~ ~ (considered as a particle) A big difference in wavelengths If electron is considered as a wave (de broglie’s formula) ~~ Here there wasn’t much difference A SIMPLE HARMONIC WAVE y = A A simple harmonic wave is a sinusoidal wave mathematically sin(kx represented as ~ - w t) 2p k = ;w = 2pu l The above equation is solution of following differential equation~ 7 Where A is the maximum amplitude of the wave, x is the space coordinate, t is the time coordinate, k is the wavenumber, ϑ is the natural frequency, ω is the angular frequency (ω = 2π ϑ) and ϕ is the phase The phase velocity for a simple harmonic wave is given as ~ Vp = ω / k PARTICLE AS A WAVE  The phase velocity of a particle that behaves like a wave is given as ~  The phase velocity of a particle cannot be greater than the speed of light  A wave extends continuously, unlocalized and is infinite, if we place different people at large distances then too they can detect the wave’s presence simultaneously  A particle is localized, fixed to a position is not infinite, different people at different places cannot detect its presence simultaneously  Then how can a particle behave like a wave?  The solution is that a particle does not behave like A wave but like A GROUP OF WAVES at once  A particle exists, travels or conveys information in a wave packet consisting of the superposition of many simple harmonic waves of different frequencies  A wave packet consists of an envelope wave and a carrier wave that is inside the envelope wave  To measure the speed of such a wave, we use group velocity and phase velocity.  Group velocity is associated with the envelope wave and phase velocity is associated with the carrier wave. 8 9 LOCALIZATION  A wave packet can be localized in space by superposing infinite number of waves of different wavelengths.  The Fourier transform can give the individual wave equations from a wave packet~  y( x ,t)   A(k )sin(kx  t)dk 0  A particle is always localized in space hence, a wave packet can be associated with a particle or moving particle.  Here the particle lies somewhere in the wave packet and the probability of finding the particle is proportional to the wave packet’s amplitude.  Even if the particle is in the wave packet and moves in it, it is impossible to find the particles exact location and velocity at a particular moment.  For a particle there is probability of finding it at a given location ~  Here for a narrow wave the amplitude is large at one location and negligible elsewhere. Due to this small region one can find the position of a particle.  But the wavelength and momentum are uncertain.  So more the wave packet is localised (turned into a narrow wave or narrower wave) more will be the precision of the particles position but one loses the precision of the particle’s momentum.  An ideal wave has a certain wavelength and momentum but it stretches from + ∞ to - ∞ so the position is uncertain in this case.  Basically, agar position precisely find karlia to momentum aur wavelength nai milega aur agar momentum aur wavelength find karlia to position nai milega  Heisenberg uncertainty principle protects wave-duality – the more accurate a particle’s position the more uncertain its momentum HEISENBERG’S UNCERTAINITY PRINCIPLE: 10  Wave packet approach is like : 11  The wave packet has to be localized in time also.  An atom spends a very short time in the excited state before spontaneous emission.  Hence, the time spread of the emitted wave packet cannot be larger than the lifetime of the state, causing uncertainty in the energy. Refer above picture. 12