What would be the normalized harmonic oscillator wavefunction correlated to the v=3 state? Show that it is also an eigenfunction of the hamiltonian operator.
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What would be the normalized harmonic oscillator wavefunction correlated to the v=3 state? Show that it is also an eigenfunction of the hamiltonian operator.
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- Estimate the lowest state energy of a system with Hamiltonian, H =- +V(x) where 2m V(x) = 9£x for r>0 0 for xs0 -bx Choose the Fang-Howard wavefunction w(x) = x e 2 as trial wave function.Demonstrate that the Hamiltonian operator for a particle experiencing a harmonic potential V (x) = 1/2 m*ω^2*x^2 is HermitianHow might I be able to prove that the Hamiltonian is linear in Problem 11.16? This problem is in a chapter called "Atomic transitions and Radiation." The chapter is under Quantum Mechanics.
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- 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)Spin/Field Hamiltonian Consider a spin-1/2 particle with a magnetic moment µ = -e/m$ placed in a uniform magnetic field aligned along the z axis. (a) Write the Hamiltonian for this system in matrix form. (b) Verify by explicit matrix calculation that the Hamiltonian does not commute with the spin operators in the r and y directions. Comment on how this affects the expectation values of these operators.Show that the operator H = -1/2(d^2/dx^2) is hermitian, assuming that it operates on a Hilbert space of L^2 functions whose functions and derivatives vanish at x = −∞ and x = +∞