2.29 Consider a particle in one dimension bound to a fixed center by a 6-function potential of the form V(x)=-vod(x) where vo is real and positive. Find the wave function and the binding energy of the ground state. Are there excited bound states?
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Q: 7.25 With the previous problem in mind prove that dn (v) dv n₂ = n(v) + v i need clear ans
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Q: 7.25 With the previous problem in mind prove that dn (v) dv n₂ = n(v) + v
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- Fast answerProblem 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.…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 .
- 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)Problem 4.45 What is the probability that an electron in the ground state of hydro- gen will be found inside the nucleus? (a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to r = 0. Let b be the radius of the nucleus. (b) Expand your result as a power series in the small number € = 2b/a, and show that the lowest-order term is the cubic: P≈ (4/3)(b/a)³. This should be a suitable approximation, provided that b