Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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Chapter 10, Problem 10.35E
For an unbound (or “free”) particle having mass
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Consider again the system in quizzes 1 and 2, namely a particle moving in one dimension
described by the normalized wavefunction
(x) =
30
1 (а — х) for 0 a .
а
Determine the expectation value () for the particle.
Imagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In eachcase, give your reasons for accepting or rejecting each function. (i) Ψ(x)=x2; (ii) Ψ(x)=1/x; (iii) Ψ(x)=e-x^2.
10
Chapter 10 Solutions
Physical Chemistry
Ch. 10 - State the postulates of quantum mechanics...Ch. 10 - Prob. 10.2ECh. 10 - State whether the following functions are...Ch. 10 - State whether the following functions are...Ch. 10 - Prob. 10.5ECh. 10 - Prob. 10.6ECh. 10 - Evaluate the operations in parts a, b, and f in...Ch. 10 - The following operators and functions are defined:...Ch. 10 - Prob. 10.9ECh. 10 - Indicate which of these expressions yield...
Ch. 10 - Indicate which of these expressions yield an...Ch. 10 - Why is multiplying a function by a constant...Ch. 10 - Prob. 10.13ECh. 10 - Using the original definition of the momentum...Ch. 10 - Under what conditions would the operator described...Ch. 10 - A particle on a ring has a wavefunction =12eim...Ch. 10 - Calculate the uncertainty in position, x, of a...Ch. 10 - For an atom of mercury, an electron in the 1s...Ch. 10 - Classically, a hydrogen atom behaves as if it were...Ch. 10 - The largest known atom, francium, has an atomic...Ch. 10 - How is the Bohr theory of the hydrogen atom...Ch. 10 - Though not strictly equivalent, there is a similar...Ch. 10 - The uncertainty principle is related to the order...Ch. 10 - Prob. 10.24ECh. 10 - Prob. 10.25ECh. 10 - For a particle in a state having the wavefunction...Ch. 10 - Prob. 10.27ECh. 10 - A particle on a ring has a wavefunction =eim,...Ch. 10 - Prob. 10.29ECh. 10 - Prob. 10.30ECh. 10 - Prob. 10.31ECh. 10 - Normalize the following wavefunctions over the...Ch. 10 - Prob. 10.33ECh. 10 - Prob. 10.34ECh. 10 - For an unbound or free particle having mass m in...Ch. 10 - Prob. 10.36ECh. 10 - Prob. 10.37ECh. 10 - Prob. 10.38ECh. 10 - Evaluate the expression for the total energies for...Ch. 10 - Prob. 10.40ECh. 10 - Verify that the following wavefunctions are indeed...Ch. 10 - In exercise 10.41a, the wavefunction is not...Ch. 10 - Prob. 10.43ECh. 10 - Prob. 10.44ECh. 10 - Explain why n=0 is not allowed for a...Ch. 10 - Prob. 10.46ECh. 10 - Prob. 10.47ECh. 10 - Prob. 10.48ECh. 10 - Carotenes are molecules with alternating CC and...Ch. 10 - The electronic spectrum of the molecule butadiene,...Ch. 10 - Prob. 10.51ECh. 10 - Prob. 10.52ECh. 10 - Show that the normalization constants for the...Ch. 10 - Prob. 10.54ECh. 10 - Prob. 10.55ECh. 10 - An official baseball has a mass of 145g. a...Ch. 10 - Is the uncertainty principle consistent with our...Ch. 10 - Prob. 10.58ECh. 10 - Prob. 10.59ECh. 10 - Instead of x=0 to a, assume that the limits on the...Ch. 10 - In a plot of ||2, the maximum maxima in the plot...Ch. 10 - Prob. 10.62ECh. 10 - Prob. 10.63ECh. 10 - The average value of radius in a circular system,...Ch. 10 - Prob. 10.65ECh. 10 - Prob. 10.66ECh. 10 - Prob. 10.67ECh. 10 - Prob. 10.68ECh. 10 - Prob. 10.69ECh. 10 - Assume that for a particle on a ring the operator...Ch. 10 - Mathematically, the uncertainty A in some...Ch. 10 - Prob. 10.72ECh. 10 - Prob. 10.73ECh. 10 - Verify that the wavefunctions in equation 10.20...Ch. 10 - An electron is confined to a box of dimensions...Ch. 10 - a What is the ratio of energy levels having the...Ch. 10 - Consider a one-dimensional particle-in-a-box and a...Ch. 10 - Prob. 10.78ECh. 10 - Prob. 10.79ECh. 10 - Prob. 10.80ECh. 10 - Prob. 10.81ECh. 10 - What are x,y, and z for 111 of a 3-D...Ch. 10 - Prob. 10.83ECh. 10 - Prob. 10.84ECh. 10 - Prob. 10.85ECh. 10 - Prob. 10.86ECh. 10 - Prob. 10.87ECh. 10 - Prob. 10.88ECh. 10 - Substitute (x,t)=eiEt/(x) into the time-dependent...Ch. 10 - Write (x,t)=eiEt/(x) in terms of sine and cosine,...Ch. 10 - Prob. 10.91ECh. 10 - Prob. 10.92ECh. 10 - Prob. 10.93ECh. 10 - Prob. 10.95E
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- The Lyman series of spectral lines for the H atom, in the ultraviolet region, arises from transitions from higher levels to n = 1. Calculate the frequency and wavelength of the least energetic line in this series.arrow_forwardA normalized wavefunction for a particle confined between 0 and L in the x direction is ψ = (2/L)1/2 sin(πx/L). Suppose that L = 10.0 nm. Calculate the probability that the particle is (a) between x = 4.95 nm and 5.05 nm, (b) between x = 1.95 nm and 2.05 nm, (c) between x = 9.90 nm and 10.00 nm, (d) between x = 5.00 nm and 10.00 nm.arrow_forwardConsider a particle trapped in a 1D box with zero potential energy with walls at x = 0 and x = L. The general wavefunction solutions for this problem with quantum number, n, are: ψn(x) = sqrt(2/L sin(npix/L)) The corresponding energy (level) for each wavefunction solution is: En = (n^2h^2)/(8mL^2) Is the particle always uniformly distributed throughout the box? Explain your answer.arrow_forward
- Imagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In each case, give your reasons for accepting or rejecting each function. (1) Þ(x) = x²; (iv) y(x) = x 5. (ii) ¥(x) = ; (v) (x) = e-* ; (iii) µ(x) = e-x²; (vi) p(x) = sinxarrow_forwardFor the system described in Exercise E7B.1(a) (A possible wavefunction for an electron in a region of length L (i.e. from x = 0 to x = L) is sin(2πx/L). Normalize this wavefunction (to 1)), what is the probability of finding the electron between x = L/4 and x = L/2?arrow_forwardFor the system described in Exercise E7B.1(a) (A possible wavefunction for an electron in a region of length L (i.e. from x = 0 to x = L) is sin(2πx/L). Normalize this wavefunction (to 1)), what is the probability of finding the electron in the range dx at x = L/2?arrow_forward
- 5arrow_forwardThe normalized wave function for a particle in a one- dimensional box in which the potential energy is zero is (x) = /2/L sin (nTx/L), where L is the length of the box (with the left wall at x = 0). What is the probability that the particle will lie between x = 0 and x = ticle is in its n = 2 state? L/4 if the par-arrow_forwardAn electron is confined to a square well of length L. What would be the length of the box such that the zero-point energy of the electron is equal to its rest mass energy, mec2? Express your answer in terms of the parameter λC = h/mec, the ‘Compton wavelength’ of the electron.arrow_forward
- Consider a particle in state n = 4 moving in a 1D box of length I = 1.0 angstrom (Å). What is the probability of finding the particle in the two middle quarters of the box? Show detailed computation.arrow_forwardThe average value for an operator is obtained by thearrow_forward5. Determine yy for each of the following wavefunctions: (i =V-1) a) y (x) = e b) y(0) = sin 0 +i cose c) w(x) = A · e*.arrow_forward
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