Advance Quantum Mechanics

Dirac-delta function, Expectation values+ mathematical interpretation, Compatible observables, Incompatible observables, Difference between continuous spectra(unbound state) and line/discrete spectra(bound state), one example, including diagrams+ graphs.

1. Advance Quantum Mechanics Assignment + Presentation Syeda Nimra Salamat

2. What is Dirac-delta function? The Dirac delta function is defined through the equations: (1) (2) Thus the delta function has an infinite value at such that the area under the curve is unity. For an arbitrary function that is continuous at   Syeda Nimra Salamat

3. What are expectation values? Explain it with mathematical interpretation? ‘In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring.’ For the position x, the expectation value is defined as This integral can be interpreted as the average value of x that we would expect to obtain from a large number of measurements. Syeda Nimra Salamat

4. Alternatively it could be viewed as the average value of position for a large number of particles which are described by the same wavefunction. Where Is the operator of x component Since the energy of a free particle is given by and the expectation value for energy becomes for a particle in one dimension. Syeda Nimra Salamat

5. In general, the expectation value for any observable quantity is found by putting the quantum mechanical operator for that observable in the integral of the wavefunction over space: Syeda Nimra Salamat

6. Explain compatible observable? ‘When two observables of a system can have sharp values simultaneously, we say that these two observables are compatible.’ If and observable are compatible that is if there exist a simultaneous set of eigenfunction of operators F and G , then these operators must commute: Example; Momentum and kinetic energy are compatible observables.   Syeda Nimra Salamat

7. two Compatible observable in above equation Syeda Nimra Salamat

8. Explain incompatible observable? A crucial difference between classical quantities and quantum mechanical observables is that the latter may not be simultaneously measurable, a property referred to as complementarity. This is mathematically expressed by non-commutativity of the corresponding operators, to the effect that the commutator. [ ] This inequality expresses a dependence of measurement results on the order in which measurements of observables and are performed.   Syeda Nimra Salamat

9. ‘Observables corresponding to non-commutative operators are called as incompatible observables.’ Incompatible observables cannot have a complete set of common eigenfunctions. Note that there can be some simultaneous eigenvectors of and ,but not enough in number to constitute a complete basis . Example; Position and momentum are incompatible observables.   Syeda Nimra Salamat

10. Incompatible observable in above equation Syeda Nimra Salamat

11. Write the difference between? Continuous spectra: (Unbound state) 1. ‘A continuous spectrum contains all the wavelengths in a given range and generates when both adsorption and emission spectra are put together.’’ 2. It is produced by white light. 3. It is characteristic of white light. 4. There are no dark spaces between colours. Discrete/Line spectra: (Bound state) 1. ‘Discrete spectrum contains only a few wavelengths and generates either in adsorption or emission.’’ 2. It is produced by  vaporization of salt or gas in discharge tube. 3. It is characteristic of atom. 4. There are dark spaces between colours. Syeda Nimra Salamat

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13. 5. Unbound states occur in those cases where the motion of the system is not confined; a typical example is the free particle. For the potential displayed in Figure there are two energy ranges where the particle’s motion is infinite:   5. If the motion of the particle is confined to a limited region of space by potential energy so that the particle move back and forth in the region then the particle is bound. 6. T he motion of the particle is bounded between the classical turning points x1 and x2 when the particle’s energy lies between 7 . The states corresponding to this energy range are called bound states.   Syeda Nimra Salamat

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15. A particle of charge q and mass m which is moving in one dimensional harmonic potential of frequency is subject to a weak electric potential field in x-direction (a) Find the exact expression for the energy? (b) Calculate the energy to first nonzero correction and compare it with the exact result obtained in (a)? Find the exact expression for the energy? The interaction between the oscillating charge and the external electric field gives rise to a term that needs to be added to the Hamiltonian of the oscillator: First, note that the eigen energies of this Hamiltonian can be obtained exactly without resorting to any perturbative treatment. A variable change   Syeda Nimra Salamat

16. This is the Hamiltonian of a harmonic oscillator from which a constant, Type equation here. , is subtracted. The exact eigen energies can thus be easily inferred:   Syeda Nimra Salamat

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19. Thank You  Hope for the best Syeda Nimra Salamat