Discuss how the Hamiltonian operator for the hydrogen atom was constructed.
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Q: The product of the two provided equations (with Z = 1) is the ground state wave function for…
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Discuss how the Hamiltonian operator for the hydrogen atom was constructed.
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- 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.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.