Tag: Solutions
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Problem 10.3 – Griffith’s Intro to QM
Problem 10.3 Prove Equation 10.33, starting with Equation 10.32. Hint: Exploit the orthogonality of the Legendre polynomials to show that the coefficients with different values of must separately vanish. Solution: Problem 10.3 Solution (Download)
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Problem 10.1 – Griffith’s Intro to QM
Problem 10.1 Rutherford scattering. An incident particle of charge and kinetic energy scatters off a heavy stationary particle of charge (a) Derive the formula relating the impact parameter to the scattering angle. Answer: (b) Determine the differential scattering cross-section. Answer: (c) Show that the total cross-section for Rutherford scattering is infinite. Solution: Problem 10.1 Solution…
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Problem 2.22 – Griffith’s Intro to QM
Problem 2.22 Evaluate the following integrals: (a) (b) (c) Solution: Problem 2.22 Solution (Download)
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Problem 2.18 – Griffith’s Intro to QM
Problem 2.18 Find the probability current, (Problem 1.14) for the free particle wave function Equation 2.95. Which direction does the probability flow? Solution: Problem 2.18 Solution (Download)
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Problem 2.17 – Griffith’s Intro to QM
Problem 2.17 Show that and are equivalent ways of writing the same function of and determine the constants and in terms of and and vice versa. Comment: In quantum mechanics, when the exponentials represent traveling waves, and are most convenient in discussing the free particle, whereas sines and cosines correspond to standing waves, which arise…
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Problem 2.15 – Griffith’s Intro to QM
Problem 2.15 Use the recursion formula (Equation 2.85) to work out and Invoke the convention that the coefficient of the highest power of is to fix the overall constant. Solution: Problem 2.15 Solution (Download)
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Problem 1.13 – Griffith’s Intro to QM
Problem 1.13 Check your results in Problem 1.11(b) with the following “numerical experiment.” The position of the oscillator at time t is You might as well take (that sets the scale for time) and (that sets the scale for length). Make a plot of at 10,000 random times, and compare it with Hint: In Mathematica,…
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Problem 9.3 – Griffith’s Intro to QM
Problem 9.3 Use Equation 9.23 to calculate the approximate transmission probability for a particle of energy that encounters a finite square barrier of height and width Compare your answer with the exact result (Problem 2.33), to which it should reduce in the WKB regime Solution: Problem 9.3 Solution (Download)
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Problem 8.1 – Griffith’s Intro to QM
Problem 8.1 Use a gaussian trial function (Equation 8.2) to obtain the lowest upper bound you can on the ground state energy of (a) the linear potential: ; (b) the quartic potential: V(x)=\alpha x^4.$ Solution: Problem 8.1 Solution (Download)
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Problem 7.3 – Griffith’s Intro to QM
Problem 7.3 Two identical spin-zero bosons are placed in an infinite square well (Equation 2.22). They interact weakly with one another, via the potential (where is a constant with the dimensions of energy, and is the width of the well). (a) First, ignoring the interaction between the particles, find the ground state and the first…