The eigenfunction equation for the energy levels of a hydrogen atom is separable into three separate equations in the spherical polar coordinates. Show that this separation cannot be performed in Cartesian coordinates, but it may be performed in parabolic coordinates
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The eigenfunction equation for the energy levels of a hydrogen atom is separable into three separate equations in the spherical polar coordinates. Show that this separation cannot be performed in Cartesian coordinates, but it may be performed in parabolic coordinates
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- Consider a hydrogen atom and we apply a perturbation potential of V = (λ/2) m(ω^2)( z^2), where λ is a dimensionless real number such that 0 < λ << 1. Use the degenerate perturbation theory to obtain the leading order correction in eigen energies for all the 4 states with n = 2.Calculate Z for a single oscillator in an Einstein solid at a temperature T=4TE=4ϵ/kBT=4 TE=4 ϵ/kB . The value of Z isDiscuss how the Hamiltonian operator for the hydrogen atom was constructed.
- Determine the polar equation for orbit r(0) of a particle in a potential energy field given by B V(r) = where a > 0 and β ≥ 0.A hydrogen atom is located in an area where there is both a uniform magnetic field and a uniform electric field that are parallel to each other. a) write out the Hamiltonian of perturbation (ignore the spin of the electron). b) use perturbation theory in order to calculate the first order correction to the energy levels n=1,2 c) is there any degeneracy left? Compare with situations in which there is a magnetic field or only an electric field.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.]
- What does the angular momentum operator L(z)(hat) equal?The wave function for the ground state of hydrogen is given by 100(0,0) = Ae¯¯r/ª Find the constant A that will normalize this wave func- tion over all space.Find the kinetic, potential, and total mechanical energies of the hydrogen atom in the first excited level, and find the wavelength of the photon emitted in a transition from that level to the ground level.