4.8 a. Assuming that the Hamiltonian is invariant under time reversal, prove that the wave function for a spinless nondegenerate system at any given instant of time can always be chosen to be real. b. The wave function for a plane wave state at t=0 is given by a complex function e/px/. Why does this not violate time-reversal invariance?
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- Please solve all the questions in the photo.There is an electron, in a 1-d, infinitely deep square potential well with a width of d. If it is in ground state, 1. Draw the electron's wavefunction. Show the position of the walls of the potential well. 2. Explain how the probability distribution for detecting the electron at a given position differs from the wavefunction.# quantum mechanical particde in a harmonic osci lator potential has the initial wave function y,)+4,(x), where Y. and Y, are the real wavefunctions in the ground and fist exci ted state of the harmonic osciclator Hamiltonian- for Convenience we take mzhzw= 1 for the oscillator- What ở the probabilpty den sity of finding the par ticke at x at time tza?
- For the Osaillator problem, mwx2 har monit (부) (2M) y e Y, LX) = Mw 1. Use the lowering operator to find Yo(X). 2. Is your wave function normalized ? Check.Normalize the following wavefunction and solve for the coefficient A. Assume that the quantum particle is in free-space, meaning that it is free to move from x € [-, ∞]. Show all work. a. Assume: the particle is free to move from x € [-0, 00] b. Wavefunction: 4(x) = A/Bxe¬ßx²Consider the following wave function. TT X a = B sin(- (x) = E 2 π.χ. a −) + C · sin(² a. Does this function describes a particle-in-a-box acceptable wave function? Name the conditions to be fulfilled. b. Is this function an eigenfunction of the total energy operator H when H is the Hamilton operator.
- A particle is placed in the potential well of finite depth U The width a of the well is fixed in such a way that the particle has only one bound state with binding energy e Uu/2 Calculate the probabilities of finding the particle in classically allowed and classically forbidden Tegiens. 2./10/Calculate the result of the transformation of the voctor operator projection R, by rotation R, around an angle a. Hint Take the second derivative of the transformed operator with respect to a and solve the second-order "uoenboConsider a particle in the one-dimensional box with the following wave function: (x,0) = Cx(a − x) 4. Normalize this wavefunction. 5. Express (x, 0) as a superposition of eigenfunctions (x). 6. What is the probability of each of these eigenfunctions? 7. Verify that the sum of all probabilities in (6) is unity. Hint: Use 8. When the system is at 9. When the system is at 10. When the system is at 11. When the system is at 12. When the system is at 13. When the system is at 14.What is (x, t) 15. What is (2) ? dt' 16. What is (d)? dt +∞ Σ n=0 1 (2n + 1)6 (x, 0), what is (x)? (x, 0), what is (²)? (x, 0), what is (p)? (x, 0), what is (p²)? (x, 0), what is Ax? (x, 0), what is Ap? = π6 960In general, a system of quantum particles can never behave even approximately like a rigid body, but non-spherical nuclei and molecules are exceptions, and have certain rotational energy levels that are well described as states of a rigid rotator whose Hamiltonian is H 21 where I is the moment of inertia, and L^ is the angular momentum operator. a, What are the eigenvalues of the angular momentum operator? b. A rigid rotator with Hamiltonian (6) is subjected to a constant perturbation V = (Ch² /21) cos² 0 ,where C is a small constant. Find the corrections to the eigenvalues and eigenfunctions to first order in perturbation theory. c. Given the perturbation condition in (b.), estimate the maximum value of C for which this is a good approximation for the level characterized by the quantum number l
- Below is a figure that depicts the potential energy of an electron (a finite square well), as well as the energies associated with the first two wave-functions. a) Sketch the first two stationary wavefunctions (solutions to the Schrődinger equation) for an electron trapped in this fashion. Pay attention to detail! Use the two dashed lines as x-axes. U(x) E2 E1 b) If the potential energy were an infinite square well (not finite well as shown above), what would the energy of the first two allowed energy levels be (i.e., E1 and E2). Write the expressions in terms of constants and a (the width of the wellI) and then evaluate numerically for a = 6.0*1010 m. [If you don't remember the formula, you can derive it by using E = h²k²/2m, together with the condition on À = 2a/n.] c) Let's say I adjust the width of the well, a, such that E1 = 3.5 ev. In that case, calculate the wavelength (in nanometers) of a photon that would be emitted in the electron's transition from E2 to E1. [Remember: hc =…4. A particle is in the state 2 1 Y (0,0)+ V5 Y, '(0,ø) – Y (0,¢), V5 which is a superposition of the normalized eigenstates, Y;" (0,¢), of the L² operator. Calculate the value of the total angular momentum of the particle in this state. Also, calculate the expectation value of the operator L+L_ in this state.