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- Indicate 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)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?How is the Bohr theory of the hydrogen atom inconsistent with the uncertainty principle? In fact, it was this inconsistency, along with the theorys limited application to non-hydrogen-like systems, that limited Bohrs theory.
- What is the physical explanation of the difference between a particle having the 3-D rotational wavefunction 3,2 and an identical particle having the wavefunction 3,2?Find the eigenvalue of operating on the function f(x) = Asin(nx) + Bcos(mx) with the following operator: P = d²/dx2 What must be the value of the constants A, B, m and n be to make the function an eigenfunction of this operator? 1.Consider a 1D particle in a box confined between a = 0 and x = 3. The Hamiltonian for the particle inside the box is simply given by Ĥ . Consider the following normalized wavefunction 2m dz² ¥(2) = 35 (x³ – 9x). Find the expectation value for the energy of the particle inside the box. Give your 5832 final answer for the expectation value in units of (NOTE: h, not hbar!). In your work, compare the expectation value to the lowest energy state of the 1D particle in a box and comment on how the expectation value you calculated for the wavefunction ¥(x) is an example of the variational principle.
- What is the eigenvalue for the eigenfunction e^(3√3ix) ?d? A = 4x² 3. Given the operator dx? determine the value of k that makes the wave function v (x)= e** an eigenfunction of 4Which of the following statements is true? I. Every one-particle Hamiltonian operator commutes with L2 and with L₂. II. For a system of n noninteracting particles, each stationary-state wave function has the form V = V1 (91) + V2 (92) +…+¥n (In) III. The energy of a system of noninteracting particles is the sum of the energies of the individual particles, where the energy of each particle is found by solving a one- particle Schrödinger equation. II and III || O I and III OI |||
- Imagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In each case, give your reasons for accepting or rejecting each function. (1) Þ(x) = x²; (iv) y(x) = x 5. (ii) ¥(x) = ; (v) (x) = e-* ; (iii) µ(x) = e-x²; (vi) p(x) = sinx2.) The function, f(x) = 3X² - 1, is an eigenfunction of the operator, A = - (1- x)(d²/ dx²) + 2x(d /dx). Find the eigenvalue corresponding to this eigenfunction.4. Given these operators A=d/dx and B=x², can you measure the expectation values of the corresponding observables to infinite precision simultaneously?