Problem 2.11 (a) Compute (x). (p). (x2), and (p²), for the states o (Equation 2.60) and ₁ (Equation 2.63), by explicit integration. Comment: In this and other problems involving the harmonic oscillator it simplifies matters if you introduce the variable = √/mw/h.x and the constant a = (mco/xh)¹/4 (b) Check the uncertainty principle for these states. (c) Compute (7) and (V) for these states. (No new integration allowed!) Is their sum what you would expect?
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- For Problem 8.16, how do I prove the relations and give the correct expressions?Problem 2.13 A particle in the harmonic oscillator potential starts out in the state ¥ (x. 0) = A[3¥o(x)+ 4¼1(x)]. (a) Find A. (b) Construct ¥ (x, t) and |¥(x. t)P. (c) Find (x) and (p). Don't get too excited if they oscillate at the classical frequency; what would it have been had I specified ¥2(x), instead of Vi(x)? Check that Ehrenfest's theorem (Equation 1.38) holds for this wave function. (d) If you measured the energy of this particle, what values might you get, and with what probabilities?Problem 2.21 Suppose a free particle, which is initially localized in the range -a < x < a, is released at time t = 0: А, if -a < х <а, otherwise, (x, 0) = where A and a are positive real constants. 50 Chap. 2 The Time-Independent Schrödinger Equation (a) Determine A, by normalizing V. (b) Determine (k) (Equation 2.86). (c) Comment on the behavior of (k) for very small and very large values of a. How does this relate to the uncertainty principle? *Problem 2.22 A free particle has the initial wave function (x, 0) = Ae ax where A and a are constants (a is real and positive). (a) Normalize (x, 0). (b) Find V(x, t). Hint: Integrals of the form e-(ax?+bx) dx can be handled by "completing the square." Let y = Ja[x+(b/2a)], and note that (ax? + bx) = y? – (b²/4a). Answer: 1/4 e-ax?/[1+(2ihat/m)] 2a Y (x, t) = VI+ (2iħat/m) (c) Find |4(x, t)2. Express your answer in terms of the quantity w Va/[1+ (2hat/m)²]. Sketch |V|? (as a function of x) at t = 0, and again for some very large t.…
- (a) Discuss your understanding of the concepts of the symmetry of a mechanical system, a conserved quantity or quantities within the mechanical system and the relation between them. Illustrate your answer with an example, but not the example in the Lecture Notes. What is the benefit of symmetry when analysing a mechanical system? (b) Consider the Lagrangian function on R? (defined by the Cartesian coordinates (x, y)) given by 1 L m (i² – ý²) + a(y² – x²), where m and a are constants. (i) Show to first order in e (that is, ignore terms of order e? and higher), that L is invariant under the transform (x, y) + (x + €Y, Y + ex). (ii) Find the integral of motion predicted by Noether's theorem for the Lagrangian function L.1.16. Establish thermodynamically the formulae v (7)= = S and v (R), V = N. Express the pressure P of an ideal classical gas in terms of the variables and 7, and verify the μl above formulae.I solved it but I need help in two parts.For first part, How to show thev formula is a solution of ODE? Second, for the third part, how to show it is bounded because I can not integratw matrix?
- Please, I want to solve the question correctly, clearly and conciselyFor Problem 8.37, how do I find <1/r> within the integral? I think that the exponent function inside of P(r) is actually troublesome to finding what I need to find; however, I am not certain of what's really the correct procedure here.2. A simple harmonic oscillator is in the state 4 = N(Yo + λ 4₁) where λ is a real parameter, and to and ₁ are the first two orthonormal stationary states. (a) Determine the normalization constant N in terms of λ. (b) Using raising and lowering operators (see Griffiths 2.69), calculate the uncertainty Ax in terms of .
- 2.5.2 (a) (b) From the results of Exercise 2.5.1, calculate the partial derivatives of f. 6, and with respect to r, e, and . With V given by 1a rsine ap är (greatest space rate of change), use the results of part (a) to calculate V-Vy. This is an alternate derivation of the Laplacian. Note. The derivatives of the left-hand V operate on the unit vectors of the right-hand V before the unit vectors are dotted together.Problem 4.25 If electron, radius [4.138] 4πεmc2 What would be the velocity of a point on the "equator" in m /s if it were a classical solid sphere with a given angular momentum of (1/2) h? (The classical electron radius, re, is obtained by assuming that the mass of the electron can be attributed to the energy stored in its electric field with the help of Einstein's formula E = mc2). Does this model make sense? (In fact, the experimentally determined radius of the electron is much smaller than re, making this problem worse).2.4. A particle moves in an infinite cubic potential well described by: V (x1, x2) = {00 12= if 0 ≤ x1, x2 a otherwise 1/2(+1) (a) Write down the exact energy and wave-function of the ground state. (2) (b) Write down the exact energy and wavefunction of the first excited states and specify their degeneracies. Now add the following perturbation to the infinite cubic well: H' = 18(x₁-x2) (c) Calculate the ground state energy to the first order correction. (5) (d) Calculate the energy of the first order correction to the first excited degenerated state. (3) (e) Calculate the energy of the first order correction to the second non-degenerate excited state. (3) (f) Use degenerate perturbation theory to determine the first-order correction to the two initially degenerate eigenvalues (energies). (3)