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- b) Determine the commutator i.e. [θ‚΂] = (΂΂ − ΂΂x)ƒ(xyz) = ?Determine whether the function f:(-1,00) → R defined by f(x) = 3 x+1-e 3 + x is concave or convex. c (a) f: (-1,00)→ R defined by f(x) = 3√√x+1-e-4x + x.Given f(x) = x+1 and g(x)= = cos x, determine the domain of the function f(g(x)).
- State whether each of the functions (a) to (d) is quadratically integrable.(a and b are positive constants.)(a)(b)(c)1/x.(d)True or false?A function that becomes infinite at a point must not be quadratically integrable.4. Given these operators A=d/dx and B=x², can you measure the expectation values of the corresponding observables to infinite precision simultaneously?(a) If  = 3x? and B = , then show that  and ß donot commute with respect to the function f(x) = sin x. Show, if the wave function, w) = A cos(kx) + iA sin(kx) is an Eigen-function of the linear momentum operator, P and if so, what is the Eigen value. (Note: A and k are constants). (b)
- Consider the simple functions below, f(x) = x? (1) f(x) = x³ (2) f(x) = x* + 2x3 (3) f(x) = sin(x) (4) f(x) = cos(x) (5) f(x) = e* (6) f(x) = sin(x) + cos(x) (7) f(x) = cos (x) + sin?(x) (8) f(x) = cos(x) +1 (9) f(x) = sin(x) + x (10) (a) Of the ten which function(s) are odd. Indicate them by their equation number and draw a box around your answer. (b) Of the ten which function(s) are neither even nor odd. Indicate them by their equation number and draw a box around your answer.Evaluate the commutator for the operators sin(kx) and (d^2)/(dx^2). Show all math.A two-level system is in a quantum state = α₁₁ + a22, which can be represented by the vector a = {a1, a2}. We are looking for the conditions under which is eigenstate of the operator c., defined below. 1) In the expression c.ỗ, c is a vector with components {Cr, Cy, Cz} ( C; are real numbers) and ỗ is a vector with components {σx, σy,σz} (σ¿ are 2 × 2 matrices). This means that the operator c. can also be represented by a 2 × 2 matrix. Write the matrix of the operator c. knowing that 0 0x = = (₁ }) = ( 5 ) . Oy σz= 0 (6-99) 1 0 1 0 2) In matrix form, the eigenvalue equation is AX AX, where A is the matrix of the operator of interest, X the column matrix representing the eigenvector and the corresponding eigenvalue. Write the eigenvalue equa- tion that needs to be verified for the quantum state to be an eigenvector of the operator c.. = 3) Note that AX = XX ⇒ (A - I)X 0, where I is the identity matrix. (AAI)X=0 is true if and only if det(A - I) = 0. Solve this equation for the operator…
- (5) Demonstrate that the function p=-; d dx y = exp(timx) is simultaneously an eigenfunction of and p²P7F.9 In this problem you will establish the commutation relations, given in eqn 7E. 14, between the operators for the x-, y-, and z-components of angular momentum, which are defined in eqn 7F.13. In order to manipulate the operators correctly it is helpful to imagine that they are acting on some arbitrary function f: it does not matter what fis, and at the end of the proof it is simply removed. Consider [,,1,] = ,-11. Consider the effect of the first term on some arbitrary function fand evaluate A D -x dx se The next step is to multiply out the parentheses, and in doing so care needs to be taken over the order of operations. (b) Repeat the procedure for the other term in the commutator, 1,1, f. (c) Combine the results from (a) and (b) so as to evaluate l f-11f;you should find that many of the terms cancel. Confirm that the final expression you have is indeed iħl_f, where l̟ is given in eqn 7F.13. (d) The definitions in eqn 7E.13 are related to one another by4) Is the function = B sin (bx) an eigenfunction of the operator d/dx = = d/dx if B and b are constants and real numbers? Show your work in detail.