QUANTUM PHYSICS DR. N. VENKATANATHAN QUANTUM PHYSICS CLASSICAL PHYSICS Classical physics is the branch of physics based on the principles developed before the rise of relativity and quantum mechanics. It deals with the set of laws describing motion of macroscopic bodies along with their forces. It shows extremely accurate results if we look about the large objects speed which is not approaching the velocity of light “c” When the object approaches velocity of light “c” classical mechanics enhanced by the special theory of relativity (i.e.) general relativity unifies classical mechanics and special relativity. Quantum Physics Deals with microscopic bodies Speed approaches or equals to the velocity of light LIMITATIONS OF CLASSICAL PHYSICS Constancy of speed of light - "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body." Black body radiation - Absorbs and emits all type of wavelength Photoelectric effect - The photoelectric effect is the observation that metals emit electrons when light incident them. Electrons emitted in this manner can be called photoelectrons. According to classical electromagnetic theory, this effect can be attributed to the transfer of energy from the light to an electron in the metal. From this perspective, an alteration in either the amplitude or wavelength of light would induce changes in the rate of emission of electrons from the metal. Furthermore, according to this theory, a sufficiently dim light would be expected to show a lag time between the initial shining of its light and the subsequent emission of an electron. However, the experimental results did not correlate with either of the two predictions made by this theory Compton Scattering - inelastic scattering - because the wavelength of the scattered light is different from the incident radiation. The effect is important because it demonstrates that light cannot be explained purely as a wave DR. N. VENKATANATHAN 1 QUANTUM PHYSICS phenomenon. Thomson scattering, the classical theory of an electromagnetic wave scattered by charged particles, cannot explain low intensity shifts in wavelength. Quantum of radiations carrying energy as well as momentum scatters off an electron. Dual nature of electron - According to this theory, small particles like electrons when in motion possess wave properties. Spectral lines emitted by the hydrogen spectrum - the emission spectrum of atomic hydrogen are divided into a number of spectral series. Zeeman Effect - splitting a spectral line into several components in the presence of a static magnetic field. Anomalous stability of atoms and molecules - In case of atoms, electrons are revolving in the extra nuclear space. According to the electro-magnetic theory of radiations a revolving charged particle radiates energy in the form of electromagnetic radiations. If a revolving electron emits a radiation, then it continuously loses its energy and it will collapse with the nucleus. This shows the instability of atom. But experimental studies show that atoms are stable. Wave and particle duality - Classical physics can deal with wave or particle. Various experiment like interference, photo electric effect, electron diffraction shows that waves sometimes act as if they were stream of particles and stream particles sometimes act as if they were waves. But which couldn’t explained by classical physics WIEN’S LAW Wavelength of the most intense radiation is inversely proportional to the absolute temperature (0 K) of an emitting body. = 1 . = Rayleigh – Jeans Law Energy radiated by a black body/second/unit area cross section between the frequency ν and ν + dν, = 8 4 Failure of Wien’s & Rayleigh Jeans Law Both laws are based on classical mechanics. According to Classical mechanics, the DR. N. VENKATANATHAN 2 QUANTUM PHYSICS oscillating particles can have any value of frequency, hence any amount of vibration energy. (i.e.) The exchange of energy between radiation and oscillators should be a perfectly continuous process and energy of the oscillator must vary continuously. This leads to Ultraviolet catastrophe. Ultraviolet catastrophe - According to classical physics energy density of an electric field in vacuum is infinite due to the divergence of energy by the shorter wavelength modes. But if we see experimentally no divergence of energy and the total energy is finite Black body radiation Black body - an opaque and non-reflective body – contributed mainly to the breakdown of classical physics - Black-body radiation has a characteristic, continuous frequency spectrum. Thermal radiation emitted by black body has following properties, The thermal radiation is independent of the material The thermal radiation strongly depends only on the body's temperature Temperature increases – radiated energy and frequency of most intense radiation increases. Spectral radiance – energy emitted per unit time per unit area for the frequency between ν and ν + dν - () Since black body emits radiation with all possible frequency, the total radiance is, DR. N. VENKATANATHAN 3 QUANTUM PHYSICS = ∫ () 0 Black body neither reflects nor transmits any radiation, therefore appear black. On heating a black body emits all possible wavelengths. Distribution of energy in black body Spectrum Black body radiation – identical to light radiation – difference is wavelength – wave length of visible light is smaller than thermal radiation. Wavelength is function of body temperature. Conclusion from energy spectrum 1. Energy – not uniformly distributed – it is discrete. 2. For a given temperature – intensity of radiation increases with increase in wavelength – maximum for particular value of wavelength – further increase in wavelength, intensity of radiation decreases. 3. The wavelengths corresponding to the maximum energy represented by the peak of the curve shifts toward shorter wavelength as temperature increases. DR. N. VENKATANATHAN 4 QUANTUM PHYSICS Planck’s theory Matter is composed of large number of oscillating particles which vibrate with different frequency – natural frequency of the particle – its overtones. Postulates Black body contains number of harmonic oscillators at molecular dimension – vibrate with all possible frequency – natural frequency & overtones. Frequency of radiation of oscillator emitted = frequency of vibration of oscillator. Oscillator emits discrete energy, it is in the order of E = nhν; where n = 1, 2, 3, … Oscillator absorbs or emits radiation in the order of hν. (i.e.) 1hν, 2hν, 3hν, … The energy changes – takes place – discontinuously & discrete manner – always an integral multiple of a small, indivisible unit (or) packet of energy – called quanta. The quantum of energy is not fixed – directly proportional to frequency. E = hν. Theory 1. The matter is composed of a large number of oscillating particles – vibrate with different frequency. 2. Energy of oscillating particle is quantized. E = nhν 3. Oscillator emits energy, when it moves from higher quantized state to lower quantized state and absorbs energy when it moves from lower quantized state to higher quantized state. 4. It neither absorbs nor emits energy as long as it remains in the same state. 5. Emission (or) absorption – energy equals to hν – called pockets of energy. Properties of Photon 1. Photon has mass and momentum = 2 ; = ℎ ℎ ℎ = 2= = 2 2 = = ℎ ℎ = 2. Energy of a photon; E = hν This quantum value is not same for all kinds of radiation, which means ν differs for each case of radiation. DR. N. VENKATANATHAN 5 QUANTUM PHYSICS 3. Non electrical nature of photon – photons are electrically neutral – not deflected by electric and magnetic fields, hence they do not ionize. 4. Speed of photon = Speed of light 5. Similar to electron, photon also existed – like electron has ‘e’ & ‘m’ – photon has ‘h’ & ‘ν’ MATTER WAVES Light radiation possesses dual nature. Interference, diffraction & polarisation explained by – wave nature of light. Photo electric effect, Compton Effect explained by – Particle nature of light. DUAL NATURE OF LIGHT RADIATION AS WAVE AND PARTICLE In 1924, Louis de Broglie suggested, like light radiation MATTER has dual property. Matter like protons & electrons – made up of discrete particles – should exhibit wave nature. DE BROGLIE CONCEPT OF MATTER WAVES (OR) POSTULATES 1. Nature loves symmetry Nature has two manifestations – matter & radiation. Since radiation possesses dual nature, matter might also possess dual nature. 2. The close parallelism between mechanics & optics Principle of least action in Mechanics – moving particle always chooses – path for which action is less. Fermat’s principle in optics – light always chooses a path for which the transit is minimal. 3. Bohr’s theory of atomic structure According to Bohr, the stable states of electrons in the atom are governed by “integer rules”. Only phenomena involving integers in physics – modes of vibration of stretched strings – imply wave motion. These similarities suggested to de Broglie that an electron (or) any other matter particle must exhibit wave property in addition to particle property. The de Broglie wavelength Particle – undergo periodic change – give rise to matter waves (ψ) = 0 sin 20 0 = DR. N. VENKATANATHAN 6 QUANTUM PHYSICS 20 = When t = t0, = 0 sin 20 0 (1) − 2 0 = (2) 2 √1− 2 = 0 sin 20 (− 2 ) (3) 2 √1− 2 Standard equation of wav motion, 2 = sin {( ) ( − )} (4) A- Amplitude; T – time period & u – velocity of wave in x – direction Comparing eq 3 & 4, 2 = 1 20 2 √1 − 2 = = 0 (5) 2 √1− 2 − =− 2 = 2 = 2 (6) According to Einstein’s equation, = 2 ; = ℎ; (7) (8) ℎ ℎ = 2 For t = t0, ℎ0 = 0 2 0 = 0 2 ℎ DR. N. VENKATANATHAN 7 QUANTUM PHYSICS Substitute in equation (5), 0 2 = ℎ√1 − = 2 ℎ 2 2 Since = 0 2 √1− 2 The wavelength of the matter wave (λ) = = Substitute the value of “u” and “ν” = 2 2 = ℎ ℎ = ℎ Alternate method = 2 ; = ℎ; ℎ ℎ = 2 ℎ = 2 ℎ ℎ = = De Broglie wave length in term of K.E, . = 1 2 = 2 2 = ; 2 = 2 2 ; . = 2 2 2 2 1 = 2 2 = √2 Substitute in λ, = ℎ √2 Calculation of de Broglie wavelength of material particles like electrons: ℎ From the equation, = DR. N. VENKATANATHAN 8 QUANTUM PHYSICS Electrons are accelerated to various velocities, so various wave lengths – higher the electron velocity – smaller the de Broglie wave length. If velocity (v) is given to electron by accelerating it through a P.D. of “V”, then Work done = Ve This work can be converted into K.E. 1 Therefore 2 2 = 2 = 2 2 ; = √ = = √2 = ℎ ℎ = √2 If m = m0 (by ignoring relativistic consideration) = ℎ √20 PHASE VELOCITY (OR) WAVE VELOCITY Each particle of matter (like electron & proton etc.,) may be regarded as consisting of a group of waves (or) a wave pocket. Each component wave propagates with a definite velocity called wave velocity (or) phase velocity. GROUP VELOCITY When a disturbance consists of number of component waves, each travelling slightly different velocity, then the resultant velocity is called “Group Velocity”. EXPERIMENTAL STUDY OF MATTERWAVES Consider a free electron which starting from rest, falls through a P.D. of “V” volts. = = ℎ √2 6.625 10−34 √2 1.6 10−19 9.11 10−31 DR. N. VENKATANATHAN = 12.27 √ 10−10 = 12.27 √ . . 9 QUANTUM PHYSICS The wavelength of the de Broglie waves for P.D. of 100 V in vacuum is 1.227 A.U. Since the wave length comparable to the wavelength of X - rays, de Broglie waves should be diffracted by the crystals like X – rays. G. P. Thomson’s Experiment Aim: To prove electrons possess wave nature. Construction A beam of cathode rays are accelerated by means of induction coil in a discharge tube (AC). Cathode rays directed towards a positive electrode (A) and made to pass through a fine hole in the anode, so that it incident on thin gold foil (F). The thickness of the gold foil is in the order of 10-8 m. The electrons emerging out from the gold foil will incident on the fluorescent screen (S). After the visual examination is done photographic plate (P) can be inserted to make permanent record. Very high vacuum should be maintained at the portion “FP”, while small amount of air is allowed to leak into section “AC” of discharge tube. It can be done with the help of needle valve. This is to produce beam of required voltage, which in turn makes the discharge tube sufficiently soft. C A To Vacuum Pump F S P Experimental Procedure Beam of electrons of known velocity is made to fall on photographic plate after traversing thin gold foil. When photographic plate is developed, symmetrical pattern of concentric rings can be seen. DR. N. VENKATANATHAN 10 QUANTUM PHYSICS The pattern is similar to the pattern produced by the X rays, when it is passed through powdered crystal. To prove electrons got diffracted. In order to confirm that the cathode rays are involved in the formation diffraction pattern, magnetic field is applied after the gold foil. Entire pattern in the screen get shifted confirms electrons get diffracted and secondary X rays are not produced. Proves wave nature of electrons Removing the gold foil “F”, the pattern get disappeared, clearly shows the presence of the foil is essential. If the electrons are in corpuscles nature, it might be scattered after the film “F”. Thus production diffraction pattern demonstrates electrons are behaving like a wave also. Verification of de Broglie wave If high voltage in the order of 50000 V is applied to accelerate electrons, then very high speed electrons will behave as a wave. = 12.27 √ The relativistic mass correction should be done for very large values of “V”. HEISENBERG’S UNCERTAINITY PRIN CIPLE It is impossible to determine precisely and simultaneously the values of both the numbers of a pair of physical variables, which describe the motion of an atomic system. For example, the position and momentum of an electron cannot be determined accurately. DR. N. VENKATANATHAN 11 QUANTUM PHYSICS ∆. ∆ = ℎ 2 ; ∆. ∆ = ℎ 2 Such pair of variables is called as canonically conjugate variables*. It is not the statement about inaccuracy of measurement. It is arises from wave property - integral part of quantum mechanical description. *Pair of variables mathematically defined in such a way that they become Fourier transforms# duals of one another. #Mathematical operation – expresses a mathematical function of time as a function of frequency. Function of time often called as the time domain and function of frequency often called as the frequency domain. THE WAVE FUNCTION Wave on a string can be described by the displacement y(x, t). In case of sound wave in air, pressure varies in space and time P(x, t). For EM waves, E and B are varying in space and time. To characterise de Broglie waves associated with a material particle, which require a quantity that varies in space and time. That quantity is called as “Wave Function”, designated by “ψ”, which is function of co-ordinates (x, y, z) and time “t”. The displacement can be positive (or) negative. (i.e.) ψ(x, y, z, t) can be positive 0r negative. But according to uncertainty principle, finding quantum particle is a probability function, hence probability cannot be negative. (i.e.) wave function cannot be negative. Ψ (x, y, z, t) is not a direct measure of presence of particle, since it is not observable. Particle having a well-defined momentum will have infinite uncertainty in co-ordinate (position). Assume a free particle, which is moving along x – axis, having well defined momentum. Then, (, ) = [(−)] = 2 ; = 2 For all three coordinates, (, ) = [[−]] Such a particle is a non –localised particle*. For a localized particle, super pose the DR. N. VENKATANATHAN 12 QUANTUM PHYSICS waves and the wave function is represented by wave packet. *cannot represent a particle whose wave function is non – zero in an limited region of space. TIME DEPENDENT SCHRӦDINGER EQUATION In classical physics the particle and waves are described by the equations of motion, similarly in quantum mechanics, particle described by wave function (, ) and the equation of motion is given by Schrödinger. = ℎ ℎ ; = ℎ = ђ ђ = ; = 2 2 Free particle wave equation in 1D Free particle which is moving along x – axis having definite momentum can described by an infinite plane wave, (, ) = [(−)] --------- (1) = 2 2 = = = ℎ ђ ђ Substitute these values in equation 1, (, ) = ђ [ (−)] ------------ (2) Differentiate with respect to x, [ (−)] = ( ) ђ ђ = ђ Multiplying by –iђ on both sides, −ђ = On differentiating, 2 2 [ (−)] ђ = ( ) 2 ђ = − DR. N. VENKATANATHAN 2 ђ2 13 QUANTUM PHYSICS Multiplying on both sides by – ђ2 2 − 2 2 = 2 2 ђ2 2 -------------- (3) Differentiating equation (2) with respect to ‘t’ [ (−)] = −[ ђ ] ђ =− ђ Multiplying iђ on both sides, ђ = --------------------- (4) R.H.S of the equation (3) & (4) are equal, since for a classical particle, 2 = 2 Hence equating L.H.S, ђ2 2 ђ = − 2 2 ----------------- (5) The above equation is called as one dimensional time dependent Schrödinger equation. For three dimensional cases the equation becomes, ђ2 2 2 2 ђ2 2 ђ = − ( + + )= − ∇ 2 2 2 2 2 ∇2 = 2 2 2 + + 2 2 2 The General wave Equation The energy ‘E’ is called as Hamiltonian (total energy of the system) of a particle in a potential V(r, t) = 2 2 + (, )----------- (7) Substituting E, p and V(r,t), DR. N. VENKATANATHAN 14 QUANTUM PHYSICS ђ = − [− ђ2 2 ∇2 + (, )] (, ) ----------------- (8) Time – dependent Schrödinger equation for a particle moving in a potential V (r, t). TIME – INDEPENDENT SCHRÖDINGER EQUATION Potential ‘V’ is a function of co-ordinates only. We can separate the variables and can obtain two equations. One equation depending on variable ‘t’ only and the other equation on variable ‘r’, (, ) = ()()----------- (9) Ψ(r) – function of space coordinate & ϕ (t) function of time Substitute equation (9) in (8), (). ђ () ђ2 = () [− 2 ∇2 + ()] () Dividing both sides by ψϕ, 1 ђ = 1 ђ2 [− 2 ∇2 + ()] () ⇓ Function of t only ⇓ Function of r only This is possible only if both sides are equal to a constant “E”. ђ () = () ђ2 2 −[ ∇ + ()] () = () 2 This called time independent Schrödinger equation. HAMILTONIAN −[ ђ2 2 ∇ + ()] () = () 2 ђ2 − [2 ∇2 + ()] --------- Hamiltonian (H) () = () DR. N. VENKATANATHAN 15 QUANTUM PHYSICS The function cannot be cancelled out of this equation, since “H” is not a simple scalar multiplier, while “E” is the value of energy. (Operator H) acting on function ψ = (total energy) multiplying with function ψ. PROBABILITY INTERPRETATION OF THE WAVE FUNCTION Wave function ψ(r, t) is not observable, but ψ must in some way be the index of presence of the particle at (r, t). Statistical interpretation is now universally accepted. ∗ = ||2 Probability density P(r, t), (, ) = |(, )|2 Probability of finding the system in volume element dτ, ||2 This quantity when integrated from –α to +α, then the total probability is equal to unity. 2 ∫|| = 1 − If the probability of finding a particle in some region increases with time, the probability of finding it outside this region decreases by the same amount. Physical significance of the wave function ψ 1. The wave function ψ is a complex quantity, cannot be measured – not observable. 2. The wave function relates the particle nature and wave nature of a matter statistically. 3. The square of the wave function is determined by multiplying the wave function by its complex conjugate. ||2 = ∗ = real quantity. 4. ||2 is the probability of finding the particle in the state, and it is a measure of position probability density (P). = ||2 = ∗ . 5. The probability of finding a particle in a volume = is = ∭||2 6. Since “P” is the probability of finding the particle, the value lies between 0 and 1. 7. For the presence of the particle, P = 1 and for the absence, P = 0. DR. N. VENKATANATHAN 16 QUANTUM PHYSICS HAMILTONIAN OPERATOR The Hamiltonian of a particle is its total energy. Hamiltonian of a particle of mass “m” moving in a potential V(r), . . () = 2 2 ---------------- (1) The operator for p is -iђ∇. Replacing ‘p’ by its operator in above equation, = (−ђ∇). (−ђ∇) ђ2 2 = − ∇ 2 2 . . ( ) = () ђ2 2 = + = − ∇ + () 2 = − ђ2 2 ∇ + () 2 EIGEN VALUES AND EIGEN FUNCTIONS OF AN OPERATOR Operator ‘A’ operating on a function () simply multiplies () by a constant , () = () () is an Eigen function of the operator ‘A’ and Eigen value 5 = 5 5 5 − Eigen function of 5 – Eigen value PARTICLE IN A 1D INFINITE SQUARE WELL Consider the solution of the time independent Schrödinger equation, for a particle (e.g. electron) moving in a 1D square well of infinite depth. The particle is bouncing back and forth between the walls of the box. The box has insurmountable potential barriers as x = 0 and x = a. i.e. the box is supposed to have walls of infinite height at x = 0 and x = a. The particle has a mass ‘m’ and its position ‘x’ at any instant is given by 0<x<a. α DR. N. VENKATANATHAN 17 QUANTUM PHYSICS ‘a’ V 0 a x The potential energy V of the particle is infinite on both sides of the box. The potential energy V of the particle can be assumed to be zero between x = 0 and x = a. The potential V(x) is given by, 0 () = { 0<< ≤ 0 ≥ The Schrödinger equation for 1D case, − ђ2 2 () = ( − )() 2 2 In the regions x<0 and x>a (outside the box), the Schrödinger equation ђ2 2 − = ( − )() = −() 2 2 () = 1 ђ2 2 =0 2 2 The wave function ψ(x) = 0 for the particle being outside the box. DR. N. VENKATANATHAN 18 QUANTUM PHYSICS Energy Eigen values and Eigen Functions Consider the region 0 ≤ x ≤ a, (i.e.) for particle inside the box, then the time independent equation, ђ2 2 − = ∵ = 0, ℎ 2 2 Rearranging the equation, 2 2 = − 2 ∵ 2 = 2 ђ2 --------- (1) The solution for this equation is, = sin + ------------ (2) To determine the constants we have to use boundary conditions = 0 = 0 = = 0 = 0 = 0 cos = cos 0 = 1 ℎ ℎ = sin The condition ψ = 0 at x = a gives = 0 But A cannot be zero, since it is constant, which would make the function zero everywhere. ∴ sin = 0 () = ℎ = 1, 2, 3, … --------------- (3) The value n = 0 is left out since it leads to ψ = 0. From the equation 3 & 1, = 2 & 2 = ђ2 2 2 2 = 2 ђ2 = 2 2 ђ2 22 ; = 1, 2, 3, … ------------------ (4) These are the energy Eigen values for the particle in the 1D box. With the above values of k, the wave function ψ becomes, = sin DR. N. VENKATANATHAN = 1, 2, 3, … 19 QUANTUM PHYSICS For each value of ‘n’, there is energy level and corresponding wave function. Each value of En is Eigen value and corresponding is Eigen function. Inside the box, the particle can only have discrete energy and corresponding energy levels, it cannot be zero. Use of normalization condition gives, ||2 = ||2 ∫ sin2 0 ||2 . = 1 =1 2 = √ 2 2 Therefore the normalised wave functions of the particle () = √ sin ; = 1, 2, 3, … The normalised wave functions 1 , 2 3 . With the help of simple particle-in-a-box problem, some important principles are revealed: 1. The energy comes out quantized. 2. The highest probability of finding the particle in the box is where the antinodes of the sine function appear. DR. N. VENKATANATHAN 20 QUANTUM PHYSICS 3. The concept of zero-point energy, which states that the energy of the system would approach E1(x) if it were cooled to absolute zero. This implies that the particle would still be able to move throughout the box (a contradiction classically). Zero-point energy is the lowest possible energy that a quantum mechanical physical system, it is the energy of its ground state. All quantum mechanical systems undergo fluctuations even in their ground state and have associated zero-point energy, a consequence of their wave-like nature. Due to the uncertainty principle every physical system to have a zero-point energy greater than the minimum of its classical potential well, even at absolute zero. For example, liquid helium does not freeze under atmospheric pressure at any temperature because of its zero-point energy. QUANTUM TUNNELLING Quantum tunnelling refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not overcome. This plays an essential role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the Sun. It has important applications to modern devices such as the tunnel diode, quantum computing, and the scanning tunnelling microscope. Tunnelling is often explained using the Heisenberg uncertainty principle and the wave–particle duality of matter. 0 0≤≥ 0 < 0 > Consider a particle of mass m and energy E < Vo incident on the barrier from left. Classically, all the particles will be reflected at x = 0, if E< Vo and all will be transmitted into the region x > a, if E > Vo. () = { DR. N. VENKATANATHAN 21 QUANTUM PHYSICS However quantum mechanically there will be finite probability for the particles to be in “region 3” even if E < Vo. This is called as barrier penetration (or) quantum tunnelling. This is possible due to the wave nature of the material particles. Particles tunnels through the region 0<x>a, never be observed as its kinetic energy is negative in that region. (Like roller coaster) The solutions of Schrödinger equation for the particle give the wave function for the 3 regions. Region 1: Wave function corresponds to two waves. One is travelling from left to right (i.e.) incident, the other from right to left (reflected by the barrier). Region 2: Two solutions are possible, one is an exponentially increasing function and the other one is an exponentially decreasing function. Region 3: The wave function is finite, which is travelling only from left to right. Thus quantum mechanically there is a probability for the particle to be in the “region 3”. For that the particle should have the solution, which is exponentially increasing function. 2 Region 1: 21 + Region 2: Region 3: 2 2 2 2 3 2 + + 2 ђ2 1 = 0 ( = 0) 2(−) ђ2 2 ђ2 2 = 0 ( = ) 3 = 0 ( = 0) 2 2( − ) = 2 & = 2 2 ђ ђ2 Therefore, Region 1: 2 1 2 + 2 1 = 0 DR. N. VENKATANATHAN 22 QUANTUM PHYSICS Region 2: Region 3: 2 2 2 2 3 2 − 2 2 = 0 + 2 3 = 0 The solutions of these equations are, 1: 1 = + − 2: 2 = − + 2: 3 = + − Where A, B, C, D, F & G are amplitude of the corresponding components of each wave. A – amplitude of wave – incident on the barrier from the left. B – Amplitude of the reflected wave in region 1. F – Amplitude of the wave, penetrating the barrier in region 2. G – Amplitude of the reflected wave in region 2. C – Amplitude of the transmitted wave in region 3. D – Amplitude of the reflected wave (non-existent) in region 3. Probability density associated with a wave function is proportional to the square of the amplitude of that function. ||2 The barrier transmission coefficient, = ||2 The reflection coefficient for the barrier surface at x = 0, = ||2 ||2 If a particle with energy E is incident on a thin energy barrier of height greater than E, there is finite probability of the particle penetrating the barrier, called the “tunneling effect”. To explain quantum tunneling following phenomena in physics can be used. 1. The emission of α – particles from radioactive nuclei. A potential well - The alpha decay of polonium-212 is the one that releases the most important quantity of energy, 8.95 MeV. It is impossible for an alpha particle to go from inside the nucleus in A to outside in B. It finds itself trapped at the bottom of a "well" as shown by the curve that represents the potential energy of interaction between the particle and the rest of the nucleus. To get from A to B, the particle must DR. N. VENKATANATHAN 23 QUANTUM PHYSICS pass through a restricted area where its kinetic energy would be negative. The permitted areas are the well where nuclear attraction dominates, and outside the well where the repulsion due to the nuclear charge prevails. The quantum tunneling or “tunnel effect” describes the fact that a particle behaves as both a particle and a wave in the infinitesimally small world where quantum mechanics replaces classical mechanics. The wave associated with an alpha particle trapped inside a nucleus has been superimposed to the previous figure. We see that the wave extends slightly outside the nucleus, where the oscillation amplitude has been amplified to make them visible. The square of the amplitude of the oscillations is, in quantum mechanics, the probability of observing the particle at a given position. So there is a probability of observing the alpha particle outside the nucleus, that is to say decay. 2. Periodic inversion of ammonia molecule – atomic clock constructed on the basis. The two energetically equivalent states of ammonia, NH3, can exchange more readily than is classically predicted - this is because the molecule can tunnel through the potential DR. N. VENKATANATHAN 24 QUANTUM PHYSICS barrier at lower energies than are required to pass through the transition state. 3. Electronic devices – tunneling diode – Josephson junction 4. Tunneling in Smell Receptors - Until quite recently, it was believed that chemical receptors in the nose (400 different kinds in humans) detected the presence of various chemicals by a lock-and-key process, which identified the molecule's physical shape. There are some issues with this theory, however. For example, ethanol and ethanethiol, which have very similar shapes, smell completely different (ethanol is the alcohol we drink - ethanethiol smells of rotten eggs). This suggests that some other identifying mechanism is at work. A theory which has been growing in popularity over the last decade or so is that smell receptors rely in part on quantum tunneling to identify chemicals. The receptors pump a small current across the odorant molecule, causing it to vibrate in a characteristic way. In order for the current to flow, however, the electrons must tunnel through the non-conducting gap between the cells of the receptor and the molecule. DR. N. VENKATANATHAN 25 QUANTUM PHYSICS Smell receptors can detect the differences between similarly-shaped molecules by tunneling a small current across them, causing a characteristic vibration. 5. Electron tunneling – Scanning tunneling Microscope Scanning Tunneling Microscopes - A Scanning Tunneling Microscope (STM) works by scanning a very sharp conducting probe across the surface of a material. An electrical current is passed down the tip of the probe, and tunnels across the gap into the material. As the gap gets wider or narrower, the tunneling current gets smaller or larger, respectively. Using this data, we can build an incredibly detailed picture of the surface, even to the point of resolving humps in the surface due to individual atoms. This technique has allowed leaps forward in our understanding of the physics and chemistry of surfaces. Schematic of an STM - the tunneling current varies with the distance between the tip and the atoms on the surface, allowing defects and even individual atoms to be mapped. Flash drives - Data on flash drives is stored in a network of memory cells made up of "floating-gate" transistors. These consist of two metal gates, a control gate and a floating gate. The floating gate is trapped in an insulating layer of metal oxide. A floating-gate transistor in its normal state registers a "1" in binary code. When an electron is attached to the floating gate, it becomes trapped in the oxide layer, affecting a change in the voltage of the control gate - a transistor in this state registers a "0" in binary. When data is erased from flash DR. N. VENKATANATHAN 26 QUANTUM PHYSICS memory, a strong positive charge applied to the control gate causes the trapped electron to tunnel through the insulating layer, returning the memory cell to a "1" state. Schematic diagram of a floating-gate transistor - Trapping an electron in the floating gate causes a change in the voltage across the control gate. Each transistor stores a bit of information. TIME DEPENDENT PERTURBATION THEORY In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an additional "perturbing" Hamiltonian representing a weak disturbance to the system. If the disturbance is not too large, the various physical quantities associated with the perturbed system (e.g. its energy levels and eigenstates) can be expressed as "corrections" to those of the simple system. These corrections, being small compared to the size of the quantities themselves, can be calculated using approximate methods such as asymptotic series. The complicated system can therefore be studied based on knowledge of the simpler one. For example, by adding a perturbative electric potential to the quantum mechanical model of the hydrogen atom, we can calculate the tiny shifts in the spectral lines of hydrogen caused by the presence of an electric field (the Stark effect). This is only approximate because the sum of a Coulomb potential with a linear potential is unstable although the tunneling time (decay rate) is very long. PROBLEMS 1. Calculate the energy of a photon of wavelength 2 Ao in electron volts. What is the momentum of this photon? E= ℎ P= = 6.62 × 10−34 × 3 × 108 2 × 10−10 × 1.602 × 10−19 6.62 ×10−34 2 ×10−10 = 6.198×103 eV (or) 6.198×103×1.602×10-19 = 9.93×10-16 J = 3.31 × 10-24 kg.m.s-1 2. Determine the energy values of an electron confined in a box of width 1 Ao. DR. N. VENKATANATHAN 27 QUANTUM PHYSICS 2 ×(6.62 ×10−34 ) 2 ℎ 2 2 E= 82 = 8 ×9.1 ×10−31 ×(10−10 )2 = 6 × 10-18 n2 J = 6 ×10−18 2 1.6 ×10−19 eV = 37.5 n2 eV 3. Find the lowest energy level and the momentum of an electron in one dimensional potential well of width 1 Ao. 2 ℎ 2 2 1 En = 82 = 2 2 = 2 12 ×(6.62 ×10−34 ) 2 ℎ 2 2 E1 = 82 = 8 ×9.1 ×10−31 ×(10−10 )2 = 6 × 10-18 J = 6 ×10−18 2 eV = 37.5 n2 eV 1.6 ×10−19 2 2 ℎ 2 En = 2 = 82 ℎ p = 2 = (6.62 ×10−34 ) 2 ×10−10 = 3.31 × 10-24 kg.m.s-1. 4. A particle is confined to one dimensional infinite potential well of width 0.2 × 10-9 m. It is found that when the energy of the particle is 230 eV its Eigen functions have 5 antinodes. Find the mass of the particle. Number of antinodes = energy level. Here 5 antinodes means, fifth level (i.e.) E5 E5 = 230 eV = 230 × 1.6 × 10-19 J; a = 0.2 × 10-9 m. 2 ℎ 2 52 ℎ 2 ℎ2 En =82 ; E5 = 82 = 52 × 82 = 25 E1 E1 = 525 = 230×1.602×10−19 25 ℎ2 = 14.7 × 10-19 J. (6.62 ×10−34 ) ℎ2 E1 =82 ; m =8E 1 2 = 8×14.7 ×10−19 ×(0.2 ×10−9 )2 = 9.3 × 10-31 kg. 2 5. A particle is moving in one dimensional potential box of infinite height of width 50Ao. Calculate the probability of finding the particle within an interval of 10Ao at the centre of the box when it is in its least energy. Given: n = 1; a = 50 × 10-10 m; x = 10 × 10-10 m. Probability of finding the particle (P) = |()2 | ∆ 2 We know ψ(x) = √ sin 2 = √ sin (Since n = 1). At the centre of the box, x = ⁄2 |()2 | = 2 2 [ (⁄2) 2 2 ] = 2 sin2 2 = 2 20 P = ∆ =50×10−10 × 10 × 10−10= 50 = 0.4 The probability of finding the particle is 0.4. 6. If a dust particle of 1 µg requires 100 s to cross a distance of 1 mm which is the separation between two rigid walls of the potential, determine the quantum numbers described by it. Mass of the particle (m) = 1 µg = 10-9 kg; t = 100 s; a = 1 mm = 10-3 m. 1 2 ℎ 2 E = 2 2 and E = 82 The particle moves 10-3 m in 100s. DR. N. VENKATANATHAN 28 QUANTUM PHYSICS Therefore the velocity of the particle in 1 second = 10−3 100 = 10−5 1 E = 2 × 10−9 × (10−5 )2 = 5 × 10−20 n2 = 82 ℎ2 = 8×10−9 ×(10−3 ) (6.62 ×10−34 )2 2 × 5 × 10−20 = 9.11×1032 Therefore quantum state (n) = 3×1016. 7. An electron is bound by a potential box of infinite height having width of 0.25 Å. Calculate the lowest three permissible energies that the electron can have. [6 eV, 24 eV, 54 eV] 8. Find the lowest energy of a neutron confined to a box 10-14 m across. [2.1 MeV] 9. Determine the wave length associated with an electron having K. E. equal to 1 MeV. [0.00726Å]. 10. A proton and α – particle have the same K. E. a) How do their speeds compare? b) How do their de Broglie wavelengths compare? c) How do their momenta compare? Given - = 4 . [a) = 2 ; ) = 1 ; ) 2 = 2 ] 11. Calculate the wavelength associated with electrons whose speed is 0.01 the speed of light. [2.4 x 10-10 m] 12. Calculate the de Broglie wavelength of an alpha particle accelerated by a P. D. of 25000 volts. [6.390 x 10-14 m] 13. The equivalent wavelength of a moving electron is 0.24 x 10-10 m. What voltage is applied between two grids will bring it to rest? [2616 V]. 14. An electron is confined to move between two rigid walls separated by 10-9 m. Find the de Broglie wavelengths representing the first three allowed energy states of the electron and the corresponding energies. {(20 Å, 10 Å, 6.7 Å); (0.38 eV, 1.52 eV, 3.42 eV)} DR. N. VENKATANATHAN 29

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# QUANTUM PHYSICS