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- Evaluate the following commutators d (a) | x, dx d (b) | х, dy d (c) dx d + x dx -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.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.
- Identify the systems for which it is essential to include a factor of 1/N! on going from Q to q : (i) a sample of carbon dioxide gas, (ii) a sample of graphite, (iii) a sample of diamond, (iv) ice.b) Determine the commutator i.e. [θ‚΂] = (΂΂ − ΂΂x)ƒ(xyz) = ?Which of the following statements describe principles or rules you should consider when writing an electron configuration? Select all that apply. Multiple answers: Multiple answers are accepted for this question Select one or more answers and submit. For keyboard navigation... SHOW MORE v a no two electrons can have the same spin in the same orbital b electrons occupy orbitals from higher to lower energy levels electrons should be divided among different orbitals within the same subshell to avoid pairing if possible d electrons occupy orbitals from lower to higher energy levels e paired electrons must have the same spin One more question about sulfur! Which of the following pictures represent orbitals in a sulfur atom in its ground state? Select all that apply. A. В. C. D. Е. B.
- (a) Consider the translational motion of a particle in a one-dimensional box of mass m, that is free to move between x = 0 and x = L: (i) State the boundary conditions that must be satisfied. (1) (ii) Draw rough sketches of three different boxes to illustrate the relationship between the energy separation and the increasing or changing length of the box. (3)Suppose that 1.0 mol of perfect gas molecules all occupy the lowest energy level of a cubic box. (a) How much work must be done to change the volume of the box by ΔV? (b) Would the work be different if the molecules all occupied a state n ≠ 1? (c) What is the relevance of this discussion to the expression for the expansion work discussed in Topic 2A? (d) Can you identify a distinction between adiabatic and isothermal expansion?6. The general expression for the density of quantum states D(e) as a function of energy & under periodic boundary conditions is D(e) = V dSe (2π)3 VkE' 7³ / = 1 where Vke is the gradient of e with respect to k, fdSe represents the integral over a surface of constant energy in k-space, and V is the volume of the system in the real space. Find the density of states for a two-dimensional quantum system with εακ
- 5) Richard Feynman called the Euler relation the most remarkable formula in mathematics. Use the Euler relation to give the value of the following quantities. eio=1 ein/2_ ein = ei2π = (Hix = (rcosx = i sin x)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…Define the term lambda max (λmax)