Discuss how the Hamiltonian operator for the hydrogen atom was constructed.
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Discuss how the Hamiltonian operator for the hydrogen atom was constructed.
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- Find the exact eigenvalues to the Hamilton-operator.2) ( Consider an electron trapped in a one-dimensional anharmonic potential. The Hamiltonian for this system is given as: H mw?£? + 2m + B&3 . Regarding the cubic term as being small, apply non- degenerate perturbation theory by letting the perturbation be: A, = ß2³. а) Calculate to first order the ground state energy of this anharmonic oscillator. b) ( | Calculate to second order the ground state energy of this anharmonic oscillator. c) ( Find the lowest order correction to the ground state wave function.The value of (p,H) the square bracket represents the poison brackets, p is the density of states and H is the hamiltonian is
- V = {0, (0 < x s a, 0sysa) {00, other values of x and y %3D Answer for the particle with mass m under the effect of the potential defined as: a) Find the energy eigenvalues and eigenfunctions of the particle using the Schrödinger Wave Equation. Most Determine the energies of the first three low-energy states. Discuss whether these situations are degenerate. b) Too small to system H' = 8xy %3D By applying Perturbation Theory in perturbation (where 8 is a small number) and 2nd order energy for the ground state considering only the first three states with the lowest energy case of calculate the correction term.Q2 (a) An oscillator consisting of a mass of 1g on a spring exhibits a period of 1 s. The velocity of the mass when it crosses the zero displacement position id 10 cm/s. (i) Is the oscillator possibly in an eigenstate of the Hamiltonian ? (ii) Find the approximate value of the quantum number n associated with the energy E of the oscillator. (iii) Has the zero point energy any significance here?Calculate the 2nd order energy shift to the ground state energy of the one-dimensional harmonic oscillator, when a perturbation of the form H₁ = Є · (²) is added to the original Hamiltonian Ho = p²/2m+ ½ mw²x². Take a ⇒ (ħ/mw) ¹/2, the characteristic length scale of the oscillator. The second order correction to level n is given by E(2) = Σ m#n ||| H₁|v0| |2 m E(0) - EO)
- Show that the function \[ S=S(q, \beta, t)=\frac{m \omega}{2}\left(q^{2}+\beta^{2}\right) \cot \omega t-m \omega q \beta \csc \omega t \] is a solution to the Hamilton-Jacobi equation for Hamilton's principal function for the linear harmonic oscillator with \[ H=\frac{1}{2 m}\left(p^{2}+m^{2} \omega^{2} q^{2}\right) \] Show that this function generates a correct solution to the motion of the harmonic oscillator.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.