H.W. Prove that the eigenvalue of 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.Under what conditions will a linear operator L̑ on a function space be Hermitian?If a wavefunction is normalized at time t=0, show that it remains normalized at all times.
- 11. Evaluate (r), the expectation value of r for Y,s (assume that the operator f is defined as "multiply by coordinate r).Why does (r) not equal 0.529 for Y,,? In this problem,use 4ardr = dt.Hermitian. Mathematically show that a. and c. d. are Hermitian and b. is not. Note: To show if an operator is Hermitian, show if (ølÔlµ) equals (ølÕlµ)". Remember, (ølÔlp)* = (µ\ô†lø) = S ¢ô*p*dx. (ø\Ô]µ) = S ¢*Ôµdx. a. Ô =x b. Ô= 4 dx с. dx d. Ô=x-i“ dxWrite a matrix representation for position and momentum operators on bases made of eigenstates oscillator modes.
- Describe the wave function of the free particle in terms of position and time variables.This question is in regard to Quantum mechanics Consider a one-dimensional quantum mechanical system. Show in the coordinate repre- sentation that the momentum operator P = -ih satisfies and thus it is a self-adjoint operator.In the operator eigenvalue equation, Af(x) =a f(x), which of the following statements is not true? the effect of the operator, A, on f(x) is to increase its magnitude by a factor of a Omultiples of f(x) would be eigenfunctions of the operator, A Of(x) is an eigenfunction of the operator, A the number, a, must be equal to 0 or 1 OOO O