Questions and Answers of Introduction To Quantum Mechanics

Consider the moving delta-function well:where v is the (constant) velocity of the well.(a) Show that the time-dependent Schrödinger equation admits the exact solution.where E = -mα2/2ћ2 is
(a) Show thatsatisfies the time-dependent Schrödinger equation for the harmonic oscillator potential (Equation 2.44). Here is any real constant with the dimensions of length.(b) Find |Ψ (x, t)|2,
Consider a particle of mass m in the potential(a) How many bound states are there?(b) In the highest-energy bound state, what is the probability that the particle would be found outside the well (x
For the distribution of ages in the example in Section 1.3.1:(a) Compute (j2) and (j2).(b) Determine Δj for each j, and use Equation 1.11 to compute the standard deviation.(c) Use your results in
(a) Find the standard deviation of the distribution in Example 1.2.(b) What is the probability that a photograph, selected at random, would show a distance x more than one standard deviation away
Consider the gaussian distributionwhere A, a, and λ are positive real constants. (The necessary integrals are inside the back cover.)(a) Use Equation 1.16 to determine A.(b) Find (x), (x2) and
Consider the wave functionwhere A, λ, and ω are positive real constants. (We’ll see in Chapter 2 for what potential (V) this wave function satisfies the Schrödinger equation.)(a) Normalize
What if we were interested in the distribution of momenta (p = mv), for the classical harmonic oscillator (Problem 1.11(b)).(a) Find the classical probability distribution ρ(p) (note that p ranges
Calculate d(p) / dt.This is an instance of Ehrenfest’s theorem, which asserts that expectation values obey the classical laws. d(p) 豐ㅏㅏ dt ax (1.38)
A particle of mass m has the wave functionwhere A and a are positive real constants.(a) Find A.(b) For what potential energy function, V(x), is this a solution to the Schrödinger equation?(c)
Imagine a particle of mass m and energy E in a potential well , sliding frictionlessly back and forth between the classical turning points (a and b in Figure 1.10). Classically, the probability of
Let Pab (t) be the probability of finding the particle in the range (a < x < b), at time t.(a) Show thatWhat are the units of J (x,t)? J is called the probability current because it tells you
Check your results in Problem 1.11(b) with the following “numerical experiment.” The position of the oscillator at time t is x(t) = A cos(ωt).You might as well take ω = 1 (that sets the scale
Suppose you add a constant V0 to the potential energy (by “constant” I mean independent of x as well as t). In classical mechanics this doesn’t change anything, but what about quantum
Show thatfor any two (normalizable) solutions to the Schrödinger equation (with the same V (x) ), Ψ1 and Ψ2. P d√x 4₁ 42 dx = 0 -8 iP
Suppose you wanted to describe an unstable particle, that spontaneously disintegrates with a “lifetime” τ. In that case the total probability of finding the particle somewhere should not be
Very roughly speaking, quantum mechanics is relevant when the de Broglie wavelength of the particle in question (h/p) is greater than the characteristic size of the system (d) . In thermal
A particle is represented (at time t = 0) by the wave function(a) Determine the normalization constant A.(b) What is the expectation value of x?(c) What is the expectation value of p?(d) Find the

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