Last updated on: December 6, 2007
Errata and Comments
“Finite Element and Boundary Element Applications
to Quantum Mechanics”
L. Ramdas Ram-Mohan
Oxford University Press, 2002
Email: LRRAM (at) WPI.edu
Some of the following comments are obvious, but I have listed all the comments, corrections, typos, errata that I have identified so far. I would be most grateful to the readers if they email me any comments, corrections or typos that are not listed below.– LRR
Ch. 1:
p14: Equation above (1.23) should read:
Ch. 2:
p34: Just below Eq. (2.9), replace
by
.
p41: replace n by “iel” in Eqs (2.22) and (2.23). A simpler way to make the correction is to include a sentence above Eq.(2.22) stating, “In the following equations we replace iel by n for convenience of notation.
p47:
In Equation (2.36) the right side should be .
p48: In Eq. (2.44) the functional is
p49: Eq. (2.45) should read
Ch.3:
p78: In Eq. (3.28) a sum over i is understood. This is the usual Einstein convention of repeated indices having an implicit sum over them.
p79: Fourth line below Table 3.1: The sentence should read, “…eigenvalues obtained to five significant figures is 39.”
Ch. 5:
p110: Equation (5.2), the left side does not need a subscript (I). It should read:
V(x) = ….
Correspondingly, on p115, in the second line below the header for subsection 5.4.1 we need only V, and not VI.
Again in line 14 on p117, we need only V and not VI.
p112: Equation (5.10), the second equation requires a factor 1/c2 on the right side
so that we may use mec2= 0.511x106eV in the denominator and
eVcm.
p112: Above Eq.(5.11), insert: “It is understood that in the following that we are
using a dimensionless variable x and the tilde on x will be dropped.”
p116: Line 7 from the bottom: It might be clearer if we multiply by Φβ rather than Φα and go through the arguments in the following. Clearly, the sum over α in Eq. (5.24) is just a summation over a running index.
p117: On the last line, in the equation (no number) the left side should have
.
p121: In Figure 5.2: the subscripts used in the right side of the displayed equation, for the quantity Q are interchanged: the column vector should have the entries: {A, -Q21A, -Q31A, -Q41A, -Q51A, 0}.
p127: Equation (5.37) should read:
and, correspondingly, Fig. 5.7 should have the coordinate labels (-d/2) and (+d/2) at the two sides of the quantum well.
Again, in Eq. (5.38) the last entry should be (+d/2).
p127: Figure 5.7 should have +d/2, in the figure, for the right boundary of the quantum well.
p127: Eq. (5.38) should have, for, the exponential to be
p128: Insert after Eq. (5.41): “The wavefunction appearing in Eq. (5.41) is normalized over the entire physical structure.”
p129: Eq. (5.43) should have “c1” replaced by “c3”. In addition, on line 8 from the top, replace c1 by c3.
Ch. 6:
p149: Equation (6.1) should have m*i being dependent on z and the order of the derivatives should be such that 1/m*i( z) occurs between the two derivatives d/dz acting on the wavefunction in the kinetic energy term.
Also, after Eq. (6.1), insert:
“The inplane dispersion is modified even in the one-band parabolic model in the layered heterostructures. So the inplane masses are complicated by this effect. Here we are simplifying the situation by assuming that the effective masses in the inplane direction are constant, and that their values in each layer correspond to the bulk values in the layers.”
p149: Equation (6.2) should have “f(z)”, with no tilde on z.
p150: Equation (6.3) should have the reduced energy ε to correspond to the energy ε(z) associated with the motion in the direct perpendicular to the planes.
p152: Equation (6.13) should have m*i instead of just m* on the right side.
p152: Equation(6.13) and the equation above it should have n(z) on the left side, rather than a z with a tilde on top of it.
p154: Eq.(6.20), for T > 0K in the second line of the equation, should have no \ell_0 or m_0 in the denominator.
p157: Equation (6.26) should have a tilde on top of each occurrence of d on both sides of the equation. The same change is needed in line 6 below Eq.(6.26) and the following sentence.
p157: Equation (6.27) should have a negative (-) sign on the left side in order to be consistent with Eq.(6.28) on the next page.
p158: The occurrences of d in the text in the second paragraph require a tilde on top of d.
p159: Again in Equation (6.29) and below it, we need a tilde on top of d.
p160: Again in Equation (6.30) and below it, we need a tilde on top of d.
p164-165: In Sec. 6.5.4, we need a tilde on top of d for each occurrence of d.
p164: Equation (6.33) should read:
Ch. 7:
p174: The first factor of 2 on the right side in Eqs. (7.12)-(7.15) should be replaced by a sum over the spin σ = ±1. This is to allow for spin splitting in the magnetic field. Correspondingly the energies E, En take on a spin index and should be En,σ.
p176: Sixth line from bottom of page: R0 should have B0-1/2 to read:
“This leads to
with B0 in Tesla.”
P187: In Eq. (7.27) the factor of (1/2π3) should be replaced by 1/(2π)3.
p189: The right side of Equation (7.32) has an exponent of (-2) on the quantity in the large parentheses.
p191: Remove the factor of
in the equation below (7.38), in Eq. (7.39), and in the equation in the line below it.
p192: Insert just before the equality sign in Eq. (7.41).
Ch. 9:
p218: Equation (9.5): N3 and N4 are interchanged and need to be switched.
p 237: Equation (9.45) should have a factor ½ on the right side.
(Note that for f(ξ,η) = 1, we expect the sum over w_i to equal ½ for the area of a standard unit triangle.)
Ch. 11:
The energies in the Tables 11.2-11.5 are given in atomic units (Hartree) where
0.5 Hartree = 1 Rydberg.
But the text says they are in Rydbergs instead of in Hartree units. For example, the field-free ground state has a binding energy 0.5 Hartree = 1.0 Rydberg.
The figure captions in Figs. 11.2 and 11.3 refer to B_0=2.35 x109G, This should be 4.70 x10^9G in the figures.
Note that the text and tables use γ. (Our PRA 40, 4777 (1989) paper used atomic units and B0=2.35 x109G. In our Computers in Physics paper B was in units of 4.70x109G. )
Eq. (11.3) should have ....γ m - γ +.., and not the plus sign it has.
Ch. 13:
P308: The last three equations, appearing in Equation (13.18), should have y1 instead of yi on the right side (This should be as in the first equation.)
Ch. 14:
p349: Equation (14.64) has to be realigned.
Ch. 15:
p364: Figure 15.1: replace 9 by 11 in the figure caption.
p364: replace 9 by 11 on the last line in the page.
Appendices:
P515: In line 4, replace (A.18) by (A.19).
p519: The integral expression for Gauss-Chebychev formula should have the factor in the argument to be
, with
z replacing x.
p524: At the end of Problem 3: remove
“Also, compare adaptive Gauss--Legendre quadrature with a straight application of Gauss-Hermite quadrature.”
Here Gauss Hermite is not applicable since the range of the integration is not from -∞ to +∞.
p532: Just above Sec. B3: Insert:
Thus the integral representation of δ(x) is:
p534: The restriction below (B.23) is not needed since:
P537: In Problem B.2 the potential function should not have a delta-function. The presence of the delta-function changes the units since the function has units of inverse length.
p542: In Eq. (C.6), the integral should have dx’.
p547: 5th line from the bottom, replace L by H.
p550: Two lines above Eq. (C.32), the eigenvalues should read
p582: Table D2: The value of the Bohr magneton should be
5.788..× 10-5 eV/T
