1. Quantum Mechanics : Matrix Diagonalization (Matrix Formulation) An operator A is expressed in matrix form as follows : 8 4 6 A = 4 14 4 6 4 8 With the above information, diagonalize the operator A.
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- 518. In differential form, what is the kinetic energy operator equal to? Does this depend on the problem that you are solving in quantum mechanics? What is the eigenvalue spectrum for this operator?2. Quantum Mechanics Please write the solutions completely (from general formula to derivation of formula) for study purposes. Thank you. Book: Quantum Mechanics Concepts and Applications - Nouredine Zettili
- 3. Plane waves and wave packets. In class, we solved the Schrodinger equation for a "free particle" (e.g. when U(x,t) = 0). The correct[solution is (x, t) = Ae(px-Et)/ħ This represents a "plane wave" that exists for all x. However, there is a strange problem with this: if you try to normalize the wave function (determine A by integrating * for all x), you will find an inconsistency (A has to be set equal to 0?). This is because the plane wave stretches to infinity. In order to actually represent a free particle, this solution needs to be handled carefully. Explain in words (and/or diagrams) how we can construct a "wave packet" from the plane wave solution. (Hint 1: consider a superposition of plane waves for a limited range of momentum/energy. Hint 2: have a look at the brief discussion in the middle of pg. 278 and especially pg. 308-309 of the text.)Subject Quantum Mechanics. Wave function normalization and superposition of solutions. Wavel functions, ψ1 and ψ2 both normalized. Find a relationship between A and B such that the superposition Aψ1 + Bψ2 is also a normalized solution. I'm having trouble with the integral of |Aψ1 + Bψ2|2 dx. Thank you!B7