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- Indicate which of these expressions yield an eigenvalue equation, and if so indicate the eigenvalue. a ddxcos4xb d2dx2cos4x c px(sin2x3)d x(2asin2xa) e 3(4lnx2), where 3=3f ddsincos g d2d2sincosh ddtanHow many nodal planes exist for a 5d orbital? (a) 0 (b) 1 (c) 2 (d) 3Indicate which of these expressions yield eigenvalue equations, and if so indicate the eigenvalue. a ddxsinx2b d2dx2sinx2 c iddxsinx2d iddxeimx, where m is a constant e ddx(ex)f (22md2dx2+0.5)sin2x3 g ddy(ey2)
- The following operators and functions are defined: A=x()B=sin()C=1()D=10()p=4x32x2q=0.5r=45xy2s=2x3 Evaluate: a Ap b Cq c Bs d Dq e A(Cr) f A(Dq)Consider a one-dimensional particle-in-a-box and a three-dimensional particle-in-a-box that have the same dimensions. a What is the ratio of the energies of a particle having the lowest possible quantum numbers in both boxes? b Does this ratio stay the same if the quantum numbers are not the lowest possible values?Is the uncertainty principle consistent with our description of the wavefunctions of the 1D particle-in-a-box? Hint: Remember that position is not an eigenvalue operator for the particle-in-a-box wavefunctions.
- What is the eigenvalue for the eigenfunction e^(3√3ix) ?Which of the following functions are eigenfunctions of the momentum operator? a)eikx, b)eax2, c)x, d)x2, e)ax+b, d)sin(x+3a)Give the corresponding eigenvalue if appropriate.How many eigenstates of a 3D particle in a box have eigenvalue of E=38h2/(8ma2) if a=b=c? Would changing c change this number?
- If operators X and Y have a common eigenfunction, which of the following statement is TRUE? a. [X, Y] = 0 b. XY - YX = 0 c. both statements are true d. both statements are falseQ.4 (a) Write down the operator A^2 for (x. d/dx) (b) The operator (x+d/dx) has the eigen value α. Determine the corresponding wave functionConsider the three spherical harmonics (a) Y0,0, (b) Y2,–1, and (c) Y3,+3. (a) For each spherical harmonic, substitute the explicit form of the function taken from Table 7F.1 into the left-hand side of eqn 7F.8 (the Schrödinger equation for a particle on a sphere) and confirm that the function is a solution of the equation; give the corresponding eigenvalue (the energy) and show that it agrees with eqn 7F.10. (b) Likewise, show that each spherical harmonic is an eigenfunction of lˆz = (ℏ/i)(d/dϕ) and give the eigenvalue in each case.