Errata/Comments for
Computational Physics, 2nd Edition

• In Example 2.2 on p.27, on the right of the two equations for computing the air drag force, the terms (B₂/m) should read B₂. Thus the equations should have been:

F_drag,x = - B₂ v v_x,i,     F_drag,y = - B₂ v v_y,i

We appreciate the help by Jeff de Jong for finding this and a few other errors.

• On p.22, as noted in footnote 5, the definition of the drag coefficient C was changed by a factor of 2 from the first edition of this book (cf. Eq.(2.9)). This actually corrected an implicit error in that edition in presenting the measured values of C in Figure 2.6 as well as to conform better to the standard definition. On the other hand, in section 2.5 on Golf, we continued to use the old definition of C in Eq.(2.34). Thus the discussion of numerical values of C in section 2.5 corresponds to half of the values of the drag coefficient in the analogous discussion on the baseball in previous sections. This invites confusion and we regret that we had not unified the definition of C in all sections. Thank you for Jeff de Jong for raising this issue.

In addition, Charles Maguire pointed out that the caption of Figure 2.2 on p.23 quotes an incorrect value of the drag coefficient; while it states C=0.5 was used, it should have been C=1 instead according to the new definition in this edition of the book.

• On p.48, just preceeding Eq.(3.1), we state that "The parallel forces add to zero, since we assume that the string doesn't stretch or break". This is incorrect except at the highest points of the pendulum where its velocity is zero since the net parallel force must provide the needed centripetal acceleration to keep the pendulum in motion along a circular arc. We thank John Goff and William DeBuvitz for alerting us to this error.
• On p.49, 4th line from the top, "where l is the length of the string" should be changed to "where l is the length of the string and is measured in radians. (Pointed out by William DeBuvitz.)
• In Example 3.3 on p.58, there are 2 typos in the signs for the time evolution equation for . The minus sign in front of the bracket should be a plus and there needs to be a minus within the brackets in front of the term (g/l) sin _i. Thus, the corrected equation should read:

_i+1 = _i + [- (g/l) sin _i - q _i + F_D sin (_D t_i) ] t

We are grateful to Denis Donnelly for pointing this out.

• On p.98, the last row of Table 4.1 gives an incorrect value for the mass of Pluto. It should read 1.3 10²² kg. We appreciate the help by Charles Maguire on this point.
• On p.125, in the expression for the moment of inertia I just after Eq.(4.23), |r₁| and |r₂| are meant to be the distances from the center of mass of the two particles to each of the particles (and not from Saturn at the origin). This notation was at best unclear as John Goff pointed out. This part should have read

where I = m₁d₁² + m₂d₂² is the moment of inertia and the distances d₁ and d₂ are measured from the center of mass of the two particles to each particle.

• On p.131, line 6 from top and paragraph 2, line 5 and again on p.138, paragraph 3, line 2, we refer to Fig. 12.47 where we should have referred to Fig. 5.1 instead. We have no idea why this happened and since there is a totally unrelated Fig. 12.47 later on, it is a confusing error. We thank Gus Hart again for spotting this problem.
• On line 3 and the last line on p.193, references to Table 7.3 appear. The references should be Table 7.1 on the same page instead.
• In Eq. (7.11) on p.194, the equal sign (=) should be the proportionality sign (∝). This is because the first step has 4 choices on the square lattice even though all later attempts will have only 3 choices (by inhibition of the immediate back tracking).

• In Eq.(7.20) on p.196, there should be a factor of D on the right-hand side of the equation. Thank you, Geron Bindseil for bringing this to our attention.
• On p.220, The text that appear in the 2nd tertiary bullet (dark dot) of Example 7.4 should have been enclosed within a box.
• On p.221, just above Example 7.5, the reference to "the box in Example 7.8" should have been to "the box in Example 7.4". This is the text that should have appeared within a box but didn't (see above). Thanks to Bernhard Gubanka for noticing this error.
• On p.225, just after Example 7.6, the reference to the Depth-first algorithm described in "Example 7.3" should have been to "Example 7.2".
• On p.229, at the beginning of the last paragraph, "Another consequence effect of ..." should read "Another consequence of ..."
• On p.265, just after Eq.(8.32), the equation defining t that appears in Eq.(8.32) should have read t 1 - zJ/k_BT (T-T_c)/T_c. In other words, the equality should have been an approximate equality since we have replaced T in the denominator by T_c where T is near T_c. This has been pointed out by John Goff. (Please note that we have defined t here as the coefficient of the term linear in m in Eq.(8.32) that came from Eq.(8.31), and thus did not define it as (T-T_c)/T_c as might be more commonly done. Of course, the two expressions are nearly equal near the critical point anyway.)
• On p.274, in the 3rd line from bottom, the reduced unit of time for Ar is given as 1.810^-12 sec, but it should have been 2.210^-12 sec instead. This is one of the many errors pointed out by Bernhard Gubanka.
• On p.280, in Eq.(9.9), v²/k_BT should be replaced by v/k_BT. The prefactor of the exponential in d-dimensions generally is proportional to v^d-1 (k_BT)^-d/2. Also, Eq.(9.10) is the correct distribution of the x-component of the velocity in any dimensions. There, you simply disregard the values of the other component(s) and just look at how v_x is distributed between negative infinity and positive infinity; then you get (9.10). We thank Charles Maguire for making us aware of the issues with these equations.
• In Eq.(9.17) on p.296, the two factors of (t)² in the denominators should both be in the respective numerator. We are grateful to Gus Hart for pointing out this error. Thus the corrected equation should read:

x_i(n+1) = 2 x_i(n) - x_i(n-1) + (t)² [ x_i+1(n) + x_i-1(n) - 2 x_i(n) ] + (t)² ( [ x_i+1(n) - x_i(n) ]³ + [x_i-1(n) - x_i(n) ]³ )

• There is a typo in the second bullet of Example 10.1 on p.311. The initial values of should be non-zero. That is, we should have ₀ = _-1 = 1, e.g., and not 0. We are grateful to Katie Sweet for pointing out this error.
• On p.317, in Fig. 10.8, we stated that the solutions from left and right match fairly well for E=-1.969. While they do match better for this energy than for E=-1.6, the error is still substantial. Much better match is obtained for about E=-1.890, and thus the latter vaule would be an acceptable solution, but not E=-1.969. Again, thank you, Bernhard Gubanka.
• On p.327, third line from the top, the reference to Eq.(10.17) should have been to Eq.(10.18) as φ is a proposed solution for the Hamiltonian (10.17) and not of it. Rather, it is a poposed solution of (10.18). Thanks to Olle Windelius for pointing this out.
• There is a typo in Eq.(10.41) on p.335 pointed out by several readers (including Trevor Byrne). The last term on the right should have a plus sign in front, not the minus. The corrected equation should read:

R(x,t+t) R(x,t) - [t/2(x)²] [ I(x+x,t+t/2) - 2 I(x,t+t/2) + I(x-x,t+t/2) ] + (t) V(x) I(x,t+t/2)

• On p.339, the calculation for the Fig. 10.17 was performed using the Crank-Nicholson method (rather than the leap-frog method) and this should have been mentioned for clarity as the stability conditions are different for the two methods.
• On p.345, there is a typo in Eq.(10.56) pointed out by Elie Kawerk. On the last line of the equation, in the numerator, the term R(x,y+x,t) should have been R(x,y+y,t). Also note, as stated at the top of p.346, we have already set x=y on that line to get the denominator to be (x)².
• On p.384, the second line of Eq.(11.31) should read:

p(i,n+1) = p(i,n) - ...

that is, the second index of p(,) on the right had side should be n instead of n-1.

• On p.440, in the second line of Eq.(12.23) the exponent -V/20 should read -V/80. That is, that line should be

_n = 0.125 e^{-V/80 .}

• This is not an error per se, but in Appendix A.3, we discuss the Verlet method and its local error of O[(t)⁴]. Gus Hart has made a rather interesting observation that the application of the Verlet method to the exactly solvable problem of the simple harmonic oscillator produces a cumulative error of O[(t)²] rahter than the naively expected one of O[(t)³]. This turns out to be true also for the radioactive decay problem. There are some arguments we could make for this unexpected behavior as a general result (under certain assumptions) due to the faster accumulation of the local errors by virtue of the way Verlet time evolution works (A.19). So the analysis has an interesting twist that neither author was aware of. This point will be further investigated and may possibly be incorporated in a future edition.
• In Appendix B.2 on p.472, there are some typos. First, on the 5th line of B.2, "x₁ < x₂ < x₃" should read "x_a < x_b < x_c". Second, in Example B.1, in the 3rd bullet, "g(x₀) g(x₁)" should read "g(x₀) g(x₁)". This error was pointed out by Jeff de Jong.
• In Appendix C.1 on p.480, lines 1 and 2, "sines of cosines" should read "sines and cosines".
• In Appendix C.4 on p.488, line 2, the refernce to Fig. A2.4 should be one to Fig. C.4 instead. This and the previous errors were pointed out by Gus Hart (as were many others).
• On p.490, there are two embarassing errors. In Eq.(C.13), the integration is over t, not . So d should be replaced by dt. In Eq.(C.14), the integral should be double, over both t and τ. So there should have been an inner (or outer) integral _- ... dt on the righthand side. These errors were discovered by Bernhard Gubanka.
• About midway down on p.498, the equation that footnote 6 should be referencing is (D.18) but it appears as (??). Thanks, Len Finegold for pointing this out.
• On p.502, in Eq.(E.9), a factor of x is missing on both terms on the right as pointed out by Dr. Anselmo.
• On p.510, lines 8 through 12 should be corrected. The original statements were not taking proper account of a factor of the panel size in multiple dimensional integration. Correctly taking account of this, the leading order cumulative error is still N^-1/d, N^-2/d, N^-4/d for the rectangular panel, trapezoidal, and Simpson’s rule integration just as in integration in one of the d-dimensions. The small errors just add in more dimensions (not multiply). Thus, these methods do not become “of no use”, but simply become less competitive against methods such as Monte Carlo integration in higher dimension, which has N^-1/2 error in any dimension.
• On p.518, in Eq.(F.4), a minus sign is missing in the exponent.
• On p.527, in the 5th line from top, "was also be" should read "can also be".
• Equations (H.4) and (H.5) in Appendix H on p.528 have misprints. The last terms on the left-hand side of these equations should be a_1Nx_N and a_2Nx_N, respectively. The "x_N" in these terms were erroneously shown as "x₁" and "x₂", respectively, in the published version.
• On p.534, just after Eq.(H.27), A x = f should be replaced by A x = b. On the same page, just after Eq.(H.30), E⁽ⁿ⁾ should be replaced by Eⁿ (i.e., E raised to the n-th power).