Planck's spectrum

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Consider the square of the derivative operator D^2 (a) Show that D^2 is a linear operator (b) Find the eigenfunctions and corresponding eigenvalues of D^2. (c) Give an example of an eigenfunction of D^2 which is not an eigenfunction of

Consider the square of the derivative operator D^2 (a) Show that D^2 is a linear operator (b) Find the eigenfunctions and corresponding eigenvalues of D^2. (c) Give an example of an eigenfunction of D^2 which is not an eigenfunction of D.

Unnormalized ground-state wavefunction of a particle

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Particle of mass m in a one-dimensional impenetrable box

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For simple harmonic oscillator H = p^2/(2m) + (K/2)x^2 turn on the potential V(x) = cx^4 Calculate the first order correction to the ground state energy.

For simple harmonic oscillator H = p^2/(2m) + (K/2)x^2 turn on the potential V(x) = cx^4 Calculate the first order correction to the ground state energy. Prove that the first order energy correction vanishes for all n = odd integer excited state levels.