These questions are 6 questions related to quantum physics: PHY256. If you could please help me with handwriting format, it is way easier to understand. No typed answers please. Thank you so much in advance!
0. Consider the ID electron tunneling problem for a rectangular barrier: (a) In the wide-barrier approximation, with E = 2 eV, Us: 3 eV and L = 10 nm, how much thinner does the barrier need to be to increase the transmission probability by a factor of 10'? (b) Show that. in order for the probability current to be nonzero. both the e \"h and 6'\" components of the wave function in must be non-vanishing in the barrier. l. A particle of mass m is in a 3D innite-potential spherical well of radius in, centered at r = 0. Assuming that the angular momentum quantum number I = 0. and that R(r) = A sintkrr satises the radial Schrodinger equation, determine both A and the discrete energy levels allowed. 2. Consider a hydrogen atom is in a superposition state: w = B I? 1112. L .. + % w; L l + % w; L _.]: where the subscripts denote an. L my. Determine the normalization constant B, and calculate the expectation value in units of h. 3. Paul Dirac once proposed a semi-classical interpretation of the Bohr magneton pl, as arising from a free electron spinning about its own axis at a constant angular velocity. (:1) Does this interpretation agree with special relativity. as far as velocity is concerned? (b) Can we x this model by modifying the free electron radius from its classical value? Hint: the classical electron radius is ('"hHJ": a solid sphere has moment of inertia flair/5. 4. A He atom can be crudely modeled by 2 electrons in a 1D innite-potential square well of width 21.. (a) Derive the eigenfunction w for the lowest~energy state with spintriplet conguration. (b) Calculate the energies of the 2 lowest-energy states with the spin-triplet conguration. Hint: For t}; to he antisymmetric under exchange. the 2 electrons must have dill'erent quantum numbers. 6 . Imagine a nanoscale turbine generator that converts rotational motion to electrical current. (a) How does angular momentum quantization affect conversion of mechanical to electrical energy? (b) What does this effect qualitatively imply about Faraday's Law of Induction? Hint: consider rotational and lield energies. and recall how magnetic ux (I) and B lield are related
