borde 9. Confirm that the angular momentum operator is hermitian. This is done by showing that: f* Audr = fÃ**dr

A system is in the state = m, an eigenstate of the angular momentum operators L² and L₂. Calculate expectation values (Lx) and (L2). You can use a faster way by physical reasoning. You can, of course use raising L, and lowering L_ operators, but it will take more time.

2. Consider a density operator p. Show that tr (p) < 1 with tr (p²) = 1 if and only if p is a pure state.

The eigenfunctions of the radial part of the Hamiltonian operator depend on both the principal and angular momentum quantum numbers, n and e. For n = 2, the possible values of are 0, 1. The radial wavefunctions in each of these cases have the form, R20 (r) = N20 2- # This function has a radial node near R₁1 (r) = N₂1 r r 4A a a This function has the opposite signs at r= ao and r= 4 ao- -r/2a -r1240 where the Nne are normalization constants in each case. (a) Plot these two wave functions and match the features indicated below with the appropriate wave function. A. R21 This function has no radial nodes (for r> 0). B.R20 (1)-2. 0

Can you elaborate on the dirac notation for the raising and lowering operators. I am not understanding how you got (6n^2+6n+3)

The commutation relations among the angular momentum operators, Lx, Ly, Lz, and β, are distinctive and characteristic of all angular momentum systems. Define two new angular momentum operators as : Îx + ily Î_ = Îx – iÎy - Î+ 2 (a) Write the operator product, ÎÎ_, in terms of β and Î₂. Note: β = Îx² + Îy² + Îz². (b) Evaluate the commutator, [1²,1+]. (L+ is shorthand for Îx ± ily.) (c) Evaluate the commutator, [L₂,L+].

A particle in a 3-dimensional quadratic box with box length L has an energy given by h² E = (n+n+n). The degeneracies of the first, second, and 8mL² third level are a. e. 1, 2, 3 1, 3, 3 b. 1, 3, 1 c. 3, 3, 3 d. 1, 2, 2

Show the relation LxL = iħL for the quantum mechanical angular momentum operator L

I'm new to Dirac notation, I know the basics of bra and kets. Howewer I can't understand this. Could you explain how the upper expresion equals below expresion. What does <x^2>0 mean? ( This is 7.36 exersixe in quantum mechanics book )

6. In Dirac notation, after the equation Bø) = b|p)is solved, we often write the solutions as {|Øn)} and {bn}. The name given to {b,} is it is the spectrum of the operator B . Essentially problem 5 and problem 6 are describing identical situations. What is the relationship between pn (x) and |Øn)? To answer this, give a mathematical answer and a physical interpretation of what it means. Hint: If you do not know how to answer this off the top of your head as being obvious, review my notes on Dirac notation and how vectors are used in quantum mechanics.