In all cases then, one can find n orthonormal eigenvectors (remember we required
<kk>
= 1 as an additional condition so that our amplitudes could be interpreted in terms of
probabilities). Since any vector in an n- dimensional space can be expressed as a
Quantum Mechanical Operators and Commutation C
I. Bra-Ket Notation
It is conventional to represent integrals that occur in quantum mechanics in a
notation that is independent of the number of coordinates involved. This is done because
the fundamental stru
* dq 1.dq N = < | >
* A dq 1.dq N = < | A > = < | A | >
* A dq 1.dq N = < | A > = < | A | >
(A) * dq 1.dq N = < A | >.
It is merely convention that an 'extra' vertical line (e.g., that appearing in < | A | >) is
inserted when an operator acting on the
configuration. This Zeff is then used to qualitatively estimate the relative energetic stability
by making use of the - Z2eff/n2 scaling of the energy.
2. The radial sizes of orbitals are governed by the product of
exp(-Zeff r/nao), an r l factor which ar
The Hydrogen Atom Orbitals B
In Chapter 1 and Appendix A, the angular and radial parts of the Schrdinger
equation for an electron moving in the potential of a nucleus of charge Z were obtained.
These "hydrogen-like" atomic orbitals are proper eigenstates
+
2SinCosCos2 2
2CosSin 2
2CosCosSin 2
r
r
r
r
r2Sin
Sin2
Sin2Cos
Cos2Cos2
2SinCosCos2
+
+
+ +
r
r r
r2
r2Sin
Cos2CosSin
CosSin
+
+
r2
r2Sin2
Analogous steps can be performed for
2
2
and
. Adding up the three contributions,
y2
z2
one obtains:
2
another constant.
The net result is that we now have two first-order differential equations of the eigenvalue
form:
A 2/x2 + D = ,
and
B 2/y2 = ' ,
and the solution of the original equation has been successfully subjected to separation of
variables. The t
(x,y) = (x) (y).
Inserting this ansatz into the above differential equation and then dividing by (x) (y)
produces:
A -1 2/x2 + B -1 2/y2 + C -1 -1 /x /y + D = E .
The key observations to be made are:
A. If A if independent of y, A -1 2/x2 must be independ
complex conjugate of this equation and using the Hermiticity property <k|R|1> =
<1|R|k>* (applied with k=l) gives k* = k.
The orthogonality proof begins with Rk = kk, and Rl = ll. Multiplying the
first of these on the left by <l| and the second by <k|
giv
measurement of R is made, the wavefunction is no longer ; it is now fm for those species
for which the value m is observed.
For example (this example and others included in this Appendix are also treated
more briefly in Chapter 1) , if the initial discuss
Time Independent Perturbation Theory D
Perturbation theory is used in two qualitatively different contexts in quantum
chemistry. It allows one to estimate (because perturbation theory is usually employed
through some finite order and may not even converge
However, because L2 and Lz do not commute in this hypothetical example, the states m
that are eigenfunctions of Lz will not, in general, also be eigenfunctions of L2 . Hence,
when Lz is measured and a particular value (say -1 h) is detected, the wavefunct
At this stage, the nature of each m is unknown (e.g., the 1 function can contain np1,
n'd 1, n'f 1, etc. components); all that is known is that m has m h as its Lz value.
Taking that sub-population (|Dm|2 fraction) with a particular m h value for Lz and
s
V. Operators That Commute and the Experimental Implications
Two hermitian operators that commute
[R , S] = RS - SR = 0
can be shown to possess complete sets of simultaneous eigenfunctions. That is, one can
find complete sets of functions that are eigenfun
to bring it to diagonal form. That is, the orthonormal cfw_gn functions can be unitarily
combined:
Gp = n Up,n gn
to produce new orthonormal functions cfw_Gp for which the corresponding matrix
elements cp,p' , defined by
R Gp = n Up,n R gn = n,n' Up,n cn,
The coefficients Cm tell, through |Cm|2, the probabilities (since is normalized to unity) of
observing each of the R eigenvalues m when the measurement is made. Once the
measurement is made, that sub-population of the sample on which the experiment was ru
particular experimental sample subjected to observation, that values of L2 equal to 2 h2 and
0 h2 were detected in relative amounts of 64 % and 36 % , respectively. This means that the
atom's original wavefunction could be represented as:
= 0.8 P + 0.6 S
-i / is of the form exp(ia) and an eigenfunction of -i /r is of the form exp(ibr). The
product exp(ia) exp(ibr) is an eigenfunction of both -i / and -i /r. The
corresponding eigenvalues are a and b, respectively; only these values will be observed if
meas
= r2Sin.
Hence in converting integrals from x,y,z to r, one writes as a short hand dxdydz =
r2Sindrdd .
C. Transforming Operators
In many applications, derivative operators need to be expressed in spherical
coordinates. In converting from cartesian to sph
XII. Spherical Coordinates
A. Definitions
The relationships among cartesian and spherical polar coordinates are given as
follows:
x2+y2+z2
z = rCos
r=
x = rSin Cos
z
= Cos -1
2 2 2
x +y +z
x
y = rSin Sin = Cos -1
2 2
x +y
The ranges of the polar v
-1
0
0
.
0
D=2
-1
2
1
0
0
. .
.
.
0
0
2
.
.
- 2
0
3
0
- 3
0
.
.
.
The matrix D operates on the unit vectors e 0 = (1,0,0.), e 1 = (0,1,.) etc. just
d
like dx operates on n(x), because these unit vectors are the representations of the n(x) in
d
the bas
g(x) = | f(i)>< f(i)| |g >
i
=
fi(x)fi*(x')g(x')dx'
i
The column vector of numbers ai fi*(x')g(x')dx' is called the representation of g(x) in
the fi(x) basis. Note that this vector may have an infinite number of components because
there may be an infinite
We will see later that performing a similarity transform expresses a matrix in a
different basis (coordinate system). If, for any matrix A, we call S-1 A S = A', we can
show that performing a similarity transform on a matrix equation leaves the form of th
(S S)ik =
S
ij S jk
j
= v j*(i)v j(k) = <v(i)v(k)> = ik.
j
We have therefore found S-1, and it is S . We have also proven the important theorem
stating that any Hermitian matrix M, can be diagonalized by a matrix S which is unitary
(S = S-1) and whose co
VIII. Hermitian Matrices and The Turnover Rule
The eigenvalue equation:
M|v(i)> = i|v(i)>,
which can be expressed in terms of its indices as:
Mkjvj(i)
= ivk(i),
j
is equivalent to (just take the complex conjugate):
vj*(i)Mkj*
= ivk*(i),
j
which, for a Her
where f is any function for which f( i) is defined. Examples include exp(M) , sin(M),
M1/2, and ( I-M)-1. The matrices so constructed (e.g., exp(M) = exp( i)| v(i)>< v(i)| ) are
i
proper representations of the functions of M in the sense that they give re
Let us assume that an inverse S-1 of S exists. Then multiply the above equation on
the left by S-1 to obtain
S-1 M S = S-1 S = I = .
This identity illustrates a so-called similarity transform of M using S. Since is diagonal,
we say that the similarity tra
IX. Connection Between Orthonormal Vectors and Orthonormal Functions.
For vectors as we have been dealing with, the scalar or dot product is defined as we
have seen as follows:
<v(i)v(j)> vk*(i)vk(j) .
k
For functions of one or more variable (we denote th
-1
2
The orthonormality of the (2) e-i t functions will be demonstrated explicitly
later. Before doing so however, it is useful to review both complex numbers and basis
sets.
A. Complex numbers
A complex number has a real part and an imaginary part and is