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- (b) Suppose a particle trapped in an one-dimensional box of width a with infinitely hard walls. Derive the normalized wave function from the solution of wave function? Find the probability of particle that can be found between 0.4a and 0.5a for the first excited state.Calculate the probability that for the 1s state the electron lies between r and r+dr. (1) Find the kinetic, potential and total energies of the hydrogen atorn in the 2nd excited level.
- An electron is confined in the ground state of a one-dimensional har- monic oscillator such that V((r – (x))²) = 10-10 m. Find the energy (in eV) required to excite it to its first excited state. (Hint: The virial theorem can help.](a) A quantum dot can be modelled as an electron trapped in a cubic three-dimensional infinite square well. Calculate the wavelength of the electromagnetic radiation emitted when an electron makes a transition from the third lowest energy level, E3, to the lowest energy level, E₁, in such a well. Take the sides of the cubic box to be of length L = 3.2 x 10-8 m and the electron mass to be me = 9.11 x 10-³¹ kg. for each of the E₁ and E3 energy (b) Specify the degree of degeneracy levels, explaining your reasoning.(d) Fermions are represented by Dirac spinors and obey the Dirac equation. The Dirac equation is (i", - m)=0 where, in the so-called chiral basis, the gamma matrices are: x=(-15) - where i runs over 1,2,3 and o are the Pauli spin matrices. i. In this basis, calculate the 'fifth' gamma matrix 75 = iyºy¹z²z³. ii. Determine the result of the projection operator (1+75) acting on the spinor - (3). X =
- (b) (deBroglie wave length) Determine the deBroglie wavelength (formula: p=h/2) of a grain of dust with diameter 1um, density 1kg m-3 and speed 1cm s. Compare your result with the diameter of the dust grain and the diameter of an atom. Comment?NoneDiscuss in detail (in words only) the Radial Wave Functions in Quantum Mechanics.