For the harmonic oscillator, the raising operator A and lowering operator A can be defined in...

 

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For the harmonic oscillator, the raising operator A and lowering operator A can be defined in dimensionless units (=w = m = 1) in the form and and A+ These operators follow the relationships A = The Hamiltonian can be written as = x - ip 2 A |n) = n + 1/n +1) x + ip 2 An) = nn - 1) = H = A+ A+ + 22 a) Determine the quantum number n which corresponds to the unnormalized energy eigen- function given by: (2x - 1)e-/2 b) Determine the unnormalized energy eigenfunctions for the adjacent states (n + 1) where n was found in part a). c) Determine the unnormalized energy eigenfunctions for the adjacent states (n-1) where n was found in part a). For the harmonic oscillator, the raising operator A and lowering operator A can be defined in dimensionless units (=w = m = 1) in the form and and A+ These operators follow the relationships A = The Hamiltonian can be written as = x - ip 2 A |n) = n + 1/n +1) x + ip 2 An) = nn - 1) = H = A+ A+ + 22 a) Determine the quantum number n which corresponds to the unnormalized energy eigen- function given by: (2x - 1)e-/2 b) Determine the unnormalized energy eigenfunctions for the adjacent states (n + 1) where n was found in part a). c) Determine the unnormalized energy eigenfunctions for the adjacent states (n-1) where n was found in part a).