The Hamiltonian of a two-dimensional isotropic harmonic oscillator takes the form kx² ky² H = Î + V = _ ¹² 2² 1² 8² + +

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The Hamiltonian Of A Two Dimensional Isotropic Harmonic Oscillator Takes The Form Kx Ky H I V 2 1 8 1 (81.88 KiB) Viewed 14 times

The Hamiltonian of a two-dimensional isotropic harmonic oscillator takes the form kx² ky² H = Î + V = _ ¹² 2² 1² 8² + + 2т дх22м дуг 2 2 The motion in the x and y directions is independent, therefore the important functions oscillator take the form Vn₁n2(x, y) = Vn₁ (2) Vn2 (y), where Y₁ denotes the eigenfunction of one-dimensional harmonic oscillator. The energy of a system is expressed as EN = hw(N+1), N=0,1,... -√√552₂ N=n+m2, B1=0,1,..., ng=0, 1, ... Find the exact expression for the energy of a two-dimensional harmonic oscillator with a form potential V¹ = ½ (z² + y²) + azy, where |a|<k (kinetic energy operator ↑ identical to that in the first equation Find the expression for the first non-vanishing energy correction for the ground state of a two-dimensional harmonic oscillator with potential V', assuming the operator Ĥ' as a disorder. Compare the result with the exact result obtained in point 2, determine the condition that the value must meet Using the degenerate perturbation calculus, find the expression for the first correction to energy and a zero-order wave function for the first excited state (N = 1) two-dimensional harmonic oscillator with potential V'. Whether the disorder completely abolishes degeneration? As above for the second excited state (N = 2). Is the disorder completely abolish degeneration?