Question

In: Physics

Extra problem: The basis of the l = 1 Hilbert space in the (L2, Lz) representation...

Extra problem: The basis of the l = 1 Hilbert space in the (L2, Lz) representation are Y11(θ, φ), Y10(θ, φ), Y1-1(θ, φ).

(1) Find the matrix expressions of Lx, Ly and Lz. Hint: make use of L+ and L-.

(2) Suppose the system is in a normalized state Ψ = c1Y11 +c2Y10. Find the possible values and corresponding possibilities when measuring Lz, L2, and Lx, respectively.

Solutions

Expert Solution


Related Solutions

7. Show that the dual space H' of a Hilbert space H is a Hilbert space...
7. Show that the dual space H' of a Hilbert space H is a Hilbert space with inner product (', ')1 defined by (f .. fV)1 = (z, v)= (v, z), where f.(x) = (x, z), etc.
1. Let ? be a finite dimensional vector space with basis {?1,...,??} and ? ∈ L(?)....
1. Let ? be a finite dimensional vector space with basis {?1,...,??} and ? ∈ L(?). Show the following are equivalent: (a) The matrix for ? is upper triangular. (b) ?(??) ∈ Span(?1,...,??) for all ?. (c) Span(?1,...,??) is ?-invariant for all ?. please also explain for (a)->(b) why are all the c coefficients 0 for all i>k? and why T(vk) in the span of (v1,.....,vk)? i need help understanding this.
What is called a basis for a vector space? What are the extra properties you expect...
What is called a basis for a vector space? What are the extra properties you expect for a good basis?why?
Prove that a projection Pe(x) is a linear operator on a Hilbert space, ,H and the...
Prove that a projection Pe(x) is a linear operator on a Hilbert space, ,H and the norm of projection Pe(x) =1 except for trivial case when e = {0}.
Let A be a bounded linear operator on Hilbert space. Show that if R(A) is closed,...
Let A be a bounded linear operator on Hilbert space. Show that if R(A) is closed, so is R(A*)
Prove that the dual space of l^1 is l^infinity
Prove that the dual space of l^1 is l^infinity
Let V be a Hilbert space. Let f(x) = ∥x∥ for x ∈ V. Using the...
Let V be a Hilbert space. Let f(x) = ∥x∥ for x ∈ V. Using the definition of Frechet differentiation, show that ∇f(x) = x for all x ̸= 0. Furthermore, show that f(x) is not Frechet differentiable at x = 0.
(a)For what range in ν is the function f(x) = x ν in Hilbert space? (4)...
(a)For what range in ν is the function f(x) = x ν in Hilbert space? (4) (b) Why are observables represented by Hermitian operators? Explain fully. (3) (c) Why are determinate states of Q eigenfunctions of Qˆ? Explain fully. (4) (d) Comment on the essential properties of reality, orthogonality and completeness for both the cases of discrete and continuous spectra.
Suppose a short-run production function is described as Q = 2L – (1/800)L2 where L is...
Suppose a short-run production function is described as Q = 2L – (1/800)L2 where L is the number of labors used each hour. The firm’s cost of hiring (additional) labor is $20 per hour, which includes all labor costs. The finished product is sold at a constant price of $40 per unit of Q. d. Suppose that labor costs remain unchanged but that the price received per unit of output increases to $50. How many labor units (L) will the...
a.) Find a basis for the row space of matrix B. b.) Find a basis for the column space of matrix B.
For the given matrix B= 1 1 1 3 2 -2 4 3 -1 6 5 1 a.) Find a basis for the row space of matrix B. b.) Find a basis for the column space of matrix B. c.)Find a basis for the null space of matrix B. d.) Find the rank and nullity of the matrix B.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT