Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 1, Problem 1.17P
(a)
To determine
The proof that
(b)
To determine
The lifetime of the particle in terms of the imaginary part of the potential.
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Problem 3.
A pendulum is formed by suspending a mass m from the
ceiling, using a spring of unstretched length lo and spring constant k.
3.1. Using r and 0 as generalized coordinates, show that
1
L =
= 5m (i² + r²0?) + mgr cos 0 –
z* (r – lo)²
3.2. Write down the explicit equations of motion for your generalized coordinates.
Assuming a one-dimensional collision, apply the conservation of energy theorem to the following system:In the system in the initial state, cart A is launched at the speed (vi ± delta vi) towards cart B, which is stationary.In the final state system, the two carts stick together and move together.The masses of the carts are known, as well as their uncertainty.Obtain a model for vf (the final speed of the carts) and its uncertainty based on known parameters only.
Consider a collision between cart A, moving at speed (vi ± delta vi), and cart B, immobile. The masses of the carts are known, as well as their uncertainty. Friction is neglected.
Using the conservation of energy theorem, program cells to predict the speed of sliders A and B after the collision as well as its uncertainty.
Then test your model with the following values:
mA=(0.47±0.05) kg
mB=(0.47±0.06) kg
vi A=(1.9±0.02) m/s
I am confused with part (e). I don't understand the steps. How is 1.427 obtained? How is the 20t moved to the left side of the equation, since it is inside the COS() function? I am just not understanding the math. Can you step it through with an explanation at each individual step?
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Introduction To Quantum Mechanics
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- Consider the function v(1,2) =( [1s(1) 3s(2) + 3s(1) 1s(2)] [x(1) B(2) + B(1) a(2)] Which of the following statements is incorrect concerning p(1,2) ? a. W(1,2) is normalized. Ob. The function W(1,2) is symmetric with respect to the exchange of the space and the spin coordinates of the two electrons. OC. y(1,2) is an eigenfunction of the reference (or zero-order) Hamiltonian (in which the electron-electron repulsion term is ignored) of Li with eigenvalue = -5 hartree. d. The function y(1,2) is an acceptable wave function to describe the properties of one of the excited states of Lit. Oe. The function 4(1,2) is an eigenfunction of the operator S,(1,2) = S;(1) + S,(2) with eigenvalue zero.arrow_forwardIn his original paper, de Broglie suggested that E = hv and p = h/λ, which hold for electromagnetic waves, are also valid for moving particles. Use these relationships to show that the group velocity ug of a de Broglie wave group is given by dE/dp, and with the help of Eq. (1.24), verify that vg = v for a particle of velocity v.arrow_forwardEvaluate the constants using the formula to = t [1- ⅟2 (ⱱ 2/c2)2 - ⅛ (ⱱ/c)4 - ⅟16 (ⱱ /c)16 – 5/128 (ⱱ/c)8 - …….] for ⱱ = 0.8 of c.arrow_forward
- Problem 4.28. Thermodynamic properties of a system of harmonic oscillators (a) Show that for one oscillator 1 f = hw + kT In(1 – e¬Bhw), (4.130) Bhw k [ In(1 – e-9hw)], (4.131) S = eßhw – 1 1 + eßhw 1 e = hw5 (4.132) Equation (4.132) is Planck's formula for the mean energy of an oscillator at temperature T. The heat capacity is discussed in Problem 4.50.arrow_forwardCalculate the following quantities for a three-dimensional classical ideal gas: (a) (vx), (b) (v?). (c) (v*vx), (d) (vàvy), (e) ((Vx + () (v²v?). bv,)^) (b a constant),arrow_forwardthat de/dx = 0 (x). **Problem 2.25 Check the uncertainty principle for the wave function in Equation 2.129. Hint: Calculating (p2) is tricky, because the derivative of has a step discontinuity at x = 0. Use the result in Problem 2.24(b). Partial answer: (p²) = (ma/h)².arrow_forward
- Consider a collision between cart A, moving at speed, and cart B, immobile. The masses of the carts are known, as well as their uncertainty. Friction is neglected. Using the conservation of energy theorem, program cells to predict the speed of sliders A and B after the collision as well as its uncertainty. mA=(0.47±0.05) kg mB=(0.47±0.06) kg vi A=(1.9±0.02) m/sarrow_forwardProblem #1: A soldier on a map reading exercise follows the path shown in the figure below. The total trip consists of four straight line paths. (NOTE: All given distance values have uncertainties of 1 m, that is, they all have 3 significant figures even if they all end in zero without a decimal point.) (a) Determine the components of vectors A, B, C, and D then write each one in component form. Ax = Bx = Cx = Dx = te je to to A = B = = D = Ay = By = Cy= Dy = End D 200 m Start 100 m 60.0⁰ A 30.0° C 300 m B x 150 marrow_forwardCalculate the (relativistic) minimum kinetic energy of an electron confined to a carbon nucleus, which can be modeled as a sphere of radius 2.7 fm. Sketch the situation, defining all of your variablesarrow_forward
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