![4.4 Further Examples
(a)Particle in a 2-D Square or 3-D Cubic Box
• Similar to 1-D case, but (x,y) or (x,y,z).
• Solutions are now defined by 2 or 3 quantum numbers
e.g. [n,m, En,m]; [n,m,l, En,m,l].
• Wavefunctions can be represented as contour plots in 2-D
(b)Harmonic Oscillator
• Similar to particle in 1-D box, but PE V(x) = ½kx2
(c) Electron in an Atom or Molecule
3-D KE operator
PE due to electrostatic interactions between electron and all
other electrons and nuclei.
T̂
V̂](https://image.slidesharecdn.com/qmrealitypart2-250306141038-3f746e92/85/The-reality-of-Quantum-mechanics-and-stuff-part-2-11-320.jpg)
The reality of Quantum mechanics and stuff part 2
- 1. QUANTUM MECHANICS- Illusion or Reality ? Prof. D. M. Parshuramkar Dept. of Physics N. H. College, Bramhapuri
- 2. Effect of Boundary Conditions 1. x = 0 = A sin kx + B cos kx = B = 0 B = 0 = A sin kx for all x 2. x = L = A sin kL = 0 sin kL = 0 kL = n n = 1, 2, 3, … (n 0, or = 0 for all x) 0 1 A=0 ? (or = 0 for all x) sin kL = 0 ?
- 3. Allowed Wavefunctions and Energies • k is restricted to a discrete set of values: k = n/L • Allowed wavefunctions: n = A sin(nx/L) • Normalization: A = (2/L) • Allowed energies: 2 2 2 2 2 2 n 2mL π n m 2 k E 2 2 2 n 8mL n E h L x n sin ψ L 2 n
- 4. Quantum Numbers • There is a discrete energy state (En), corresponding to a discrete wavefunction (n), for each integer value of n. • Quantization – occurs due to boundary conditions and requirement for to be physically reasonable (Born interpretation). • n is a Quantum Number – labels each allowed state (n) of the system and determines its energy (En). • Knowing n, we can calculate n and En.
- 5. Properties of the Wavefunction • Wavefunctions are standing waves: • The first 5 normalized wavefunctions for the particle in the 1-D box: • Successive functions possess one more half-wave ( they have a shorter wavelength). • Nodes in the wavefunction – points at which n = 0 (excluding the ends which are constrained to be zero). • Number of nodes = (n-1) 1 0; 2 1; 3 2 … L x n sin ψ L 2 n
- 6. Curvature of the Wavefunction • If y = f(x) dy/dx = gradient of y (with respect to x). d2 y/dx2 = curvature of y. • In QM Kinetic Energy curvature of • Higher curvature (shorter ) higher KE • For the particle in the 1-D box (V=0): ..... 2 2 x ψ T KE 2 2 2 n 2 n n L n x ψ T E
- 7. Energies • En n2 /L2 En as n (more nodes in n) En as L (shorter box) n (or L) curvature of n KE En 2 2 2 n 8mL n E h L 1 2 node E L1 E L2
- 8. • En n2 energy levels get further apart as n • Zero-Point Energy (ZPE) – lowest energy of particle in box: • CM Emin = 0 • QM E = 0 corresponds to = 0 everywhere (forbidden). E n 1 2 3 0 2 2 1 8mL E h 2 2 2 8mL 4 E h 2 2 3 8mL 9 E h 2 2 8mL ZPE h 2 2 1 min 8mL E E ZPE h
- 9. • If V(x) = V 0, everywhere in box, all energies are shifted by V. V 8mL n E 2 2 2 n h E1 E2 V = 0 V E1=E1+V E2=E2+V V 0
- 10. Density Distribution of the Particle in the 1-D Box • The probability of finding the particle between x and x+dx (in the state represented by n) is: Pn(x) = n(x)2 dx = (n(x))2 dx (n is real) • Note: probability is not uniform – n 2 = 0 at walls (x = 0, L) for all n. – n 2 = 0 at nodes (where n = 0). 2 2 dx L x n sin x P 2 L 2 n
- 11. 4.4 Further Examples (a)Particle in a 2-D Square or 3-D Cubic Box • Similar to 1-D case, but (x,y) or (x,y,z). • Solutions are now defined by 2 or 3 quantum numbers e.g. [n,m, En,m]; [n,m,l, En,m,l]. • Wavefunctions can be represented as contour plots in 2-D (b)Harmonic Oscillator • Similar to particle in 1-D box, but PE V(x) = ½kx2 (c) Electron in an Atom or Molecule 3-D KE operator PE due to electrostatic interactions between electron and all other electrons and nuclei. T̂ V̂
- 12. yudh 12 A SUMMARY OF DUAL ITY OF NATURE Wave particle duality of physical objects LIGHT Wave nature -EM wave Particle nature -photons Optical microscope Interference Convert light to electric current Photo-electric effect PARTICLES Wave nature Matter waves -electron microscope Particle nature Electric current photon-electron collisions Discrete (Quantum) states of confined systems, such as atoms.
- 13. Yodh 13 QUNATUM MECHANICS: ALL PHYSICAL OBJECTS exhibit both PARTICLE AND WAVE LIKE PROPERTIES. THIS WAS THE STARTING POINT OF QUANTUM MECHANICS DEVELOPED INDEPENDENTLY BY WERNER HEISENBERG AND ERWIN SCHRODINGER. Particle properties of waves: Einstein relation: Energy of photon = h (frequency of wave). Wave properties of particles: de Broglie relation: wave length = h/(mass times velocity) Physical object described by a mathematical function called the wave function. Experiments measure the Probability of observing the object.
- 14. Yodh 14 A localized wave or wave packet: 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 A moving particle in quantum theory
- 15. Yodh 15 ILLUSTRATION OF MEASUREMENT OF ELECTRON POSITION Act of measurement influences the electron -gives it a kick and it is no longer where it was ! Essence of uncertainty principle.
- 16. Yodh 16 Classical world is Deterministic: Knowing the position and velocity of all objects at a particular time Future can be predicted using known laws of force and Newton's laws of motion. Quantum World is Probabilistic: Impossible to know position and velocity with certainty at a given time. Only probability of future state can be predicted using known laws of force and equations of quantum mechanics. Observer Observed Tied together
- 17. Yodh 17 BEFORE OBSERVATION IT IS IMPOSSIBLE TO SAY WHETHER AN OBJECT IS A WAVE OR A PARTICLE OR WHETHER IT EXISTS AT ALL !! QUANTUM MECHANICS IS A PROBABILISTIC THEORY OF NATURE UNCERTAINTY RELATIONS OF HEISENBERG ALLOW YOU TO GET AWAY WITH ANYTHING PROVIDED YOU DO IT FAST ENOUGH !! example: Bank employee withdrawing cash, using it ,but replacing it before he can be caught ... CONFINED PHYSICAL SYSTEMS – AN ATOM – CAN ONLY EXIST IN CERTAIN ALLOWED STATES ... . THEY ARE QUANTIZED
- 18. Yodh 18 COMMON SENSE VIEW OF THE WORLD IS AN APPROXIMATION OF THE UNDERLYING BASIC QUANTUM DESCRIPTION OF OUR PHYSICAL WORLD ! IN THE COPENHAGEN INTERPRETATION OF BOHR AND HEISENBERG IT IS IMPOSSIBLE IN PRINCIPLE FOR OUR WORLD TO BE DETERMINISTIC ! EINSTEIN, A FOUNDER OF QM WAS UNCOMFORTABLE WITH THIS INTERPRETATION Bohr and Einstein in discussion 1933 God does not play dice !
- 19. Einstein-Podosky-Rosen (EPR)Paradox .Quantum entanglement .Double slit Exp. :Quantum description of Nature is incomplete.
- 20. What is QM trying to tell us? Bohr: In our description of Nature , the purpose is not to disclose the real essence of the phenomena but only to track down , so far as it is possible, relations between the manifold aspects of our experience.
- 21. 19th century problem : What is Electrodynamics trying to tell us? Fields in empty space have physical reality ; the medium that supports them does not
- 22. 20th century problem : What QM is trying to tell us ? Correlations have physical reality ; that which that correlate does not. Correlation between energy states is the reality ; not the energy states.
- 23. Planck’s guiding spirit: There are absolute laws in Nature that must be simple and logical.
- 24. Concluding Remark: QM has been an unqualified success in quantitatively describing the atomic and sub-atomic world, its interpretative aspects have not been satisfactory.
- 25. THANKS..