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hw2 - 3 A 1-dimensional particle has a wavefunction ψ =...

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Homework Problem Set No. 2 – Physics 17 Due: Wednesday, October 15, in class 1. Consider an atom with a new version of the electric potential energy such that U = - K 0 R 1 . 5 where R is the distance from the nucleus which is assumed to be infintely heavy and K 0 is a new constant. Assume an electron of mass, m , in a circular orbit around the nucleus. Compute the speed, V , of the electron as a function of R . Assume that the orbit is quantized as in the Bohr atom. For the electron, what is the relationship between R and V ? What are the quantized energy levels of this system? What is the radius of an electron’s orbit in the lowest energy level? 2. Use the formula for the energy levels of the Bohr atom to find the frequency of a tran- sition between n and n - 1. An n → ∞ , show that this frequency varies as n - 3 . Compare this frequency with the 2 π/P where P is the orbital period of the electron in the n ’th state.
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Unformatted text preview: 3. A 1-dimensional particle has a wavefunction: ψ = Axe-x 2 L 2 (1) Normalize the wavefunction. That is, for a given value of L , find A such that: Z ∞-∞ ψ 2 dx = 1 (2) What are the units of L and A ? 4. Consider a hydrogen atom in its ground state. If the uncertainty in the electron’s position equals the Bohr radius, then what is the uncertainty in the electron’s momentum? What is the mean value of the momentum that you derive from the kinetic energy of the electron? How does this mean value compare with the uncertainty derived in the previous part of this problem? 5. Assume a very simple model of a diatomic molecule with two atoms of mass m 1 and m 2 separated by a distance d . What is the moment of inertia of this molecule? If the angular momentum is quantized in units of ~ what are the energy levels? 1...
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