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**Unformatted text preview: **SCHEM3541 Tutorial
ASCC Vectors and Vector Differential O erators Vectors in three dimensions Every vector v in three-dimensional space may be written as a linear combination of A ‘, j,l .
i.e. _V_=az”+bj+ck k
where i, j,k are unit vectors j
, right
1 hand
system ' The inner product (Dot or Scalar) A e.g. g = (If + a2} + aak v=b,§+b2}'+b312 0056 = (1,171 +a2b2 + (23133
124'le lui=wla3+a§+a§ ; 1v]: b12+b§+b§ 2 -f =1
For unit vectors, ] j = 1 deﬁnition of unit vector
12-12 =1
orthonormal basis
ij=o, 222:0, j12=o ...... orthogonal The vector or cross product Given the vectors y_ and y, we deﬁne the vector or cross product
3 = y_ x 2 order is important Where 111 is a vector perpendicular to a plane determined by 14 and y_ ; the magnitude of y is
w estuarine v right-handed
u triple
For unit vectors,
{ml-.0, jx}=‘0 12x12=0
2x5=12 Mb];
but fxi=z° and kAx}=—f
IEXi=j 2x129;- Simply i \ Q J _
' k or also can be written in some simple form 3‘5 PK 2 (“253 " “3(5)? +(a3b1 _ 01/53)} +(a1bz _ ‘1sz)
Derivative of a vector If HO and G(f) are differentiable vector functions, and f(t) a differentiable scalar
function, the following differentiation rules are easily obtained. Fm = mt)? mm} + not then, - F'm- — 49‘”: =1: (0? +12 (2)} +12 (0k I. g-g = 0 , if Q is a constant vector. (independent oft)
t . Since vector multiplication is not commutative, it is important to note that the order of the
vectors in V must be preserved. Partial Differentiation Deﬁnitions: If a function ﬁx) depends only on the one variable x, its derivative which is
deﬁned as df(x)_ “m {fawn—foo}
dx —Ax—>0 Ax is called an exact derivative If a function g(x, y, 2....) depends on more than one independent variable than its derivative
with respect to these variables are called Partial derivative ’ a . lim g(x +Ax,y,Z---) - sauna-J}>
. . —— x, ,z... a _______..___..______—
e g ax g( y ) Ax ——> 0{ Ax
1‘ _ I , ‘ .
or Emma”); 1m g(x,y+Ay,z...) g<x,y,z...)
6y Ay ——> O Ay
6 d . . . . . .
where 5— (rather than E) 18 a reminder that we are takmg the derivative With respect to x
x while holding all other variable ﬁxed. a2 62
axay ’ 55c? ’ High order
Partial derivative example: (i) for U (x, y, z) = x2 + 2xyz + (y - z)2 e" ﬁ=2x+2yz+(y—z)2e" 6x E’i r: 2x2 + 202 — z)" a"
5y E = 2xy—2(y—z)e“
62 Practically, taking a partial derivative is exactly the same as taking an exact derivative, as
all independent variables other than the one considered are simply treated as constants. For'higher order partial derivatives; simply repeat this procedure azu a ﬂu
. . F b ———~=——- _ =2+ _ 2 .r
eg . roma ove 6x2 6x(6x) (y z) e
a an azu a 2
“—‘= =-—2+2 + _ -‘___2+2_ x
6y(6x)~ ‘ayax ay{x yz 0’ ”6} Z (y Z)€
equal
6 Bu 62a
-—--——- : :2 +2 __ ex-
6x(6y) axay z (y z) e.g. Recall ideal gas law PV=nRT P=———-—- the effect on pressure when small changes in no. of moles, temperatures and volume, is . (11’: £ dn+ E dT+£§f~JdV
611 GT 6V JRTFLELZK]
V n T V and as long as the changes dn, dT, d V are small, this formula gives a very accurate estimate
of the exact change in pressure. The___o};_e_m_: The order of differentation in high order partial derivatives doesn’t effect the
value of the result 6214 _62u . _a__2u _a_2__u
axay “ ayax ’ ayaz =azay i.e. Total differentials The total differential of a function u(x, y, 2,...) the change in the function caused by
the small changes (11:, dy, dz, in its independent variables. Theorem: For a function u(x, y, z,...) , the total differential is PV = nRT . ———> example du _§_u dx +6u dy+ Bu dz dt axdt +y6 dt+ azdt /’ W\\ exact derivative partial derivative exact derivatives
(because indirectly because u depends because each of the
through the x,y,z..., on several variable’s variables depends on
u depends only on t) the one variable, t Chain Rule: Vector Differentiation Operators:— Simply combine differential operators such as 3,3,2— With vectors.
' 6x By 82 There exist different operators we will consider (a) The Gradient or “Del” operator is The gradient of a scalar function g(x, y, z) is a vector. VBCtOI‘ scalar function The components of this vector represent the rate of change of the function in the' three
coordinate directions. ' e.g. g(x,y,z)= x2 +2xy+2yz+y2 +z2 +3xz 17g = f(2x+2y+32)+j(25c+22+2§)+l€(2y+22+3x) (b) The Divergence of a vector function is a scalar function which is formed by taking the
dot product of the vector operator “del” ~ -a-,—a—,-a-_ ', with the vector function
"‘ 6x By 62 VCCtOI‘ div(A)= j-A=[§§—+]§+l§§—](M, +}'A2 +IEA3)
x y Z _ — -
= 6A1 + 6A2 + 6A3 6x (By-5.2—
F.._.Y_____J scalar Note: The divergence of a vector is a scalar [c.f. the gradient of a scalar ﬁmction is a
vector] (c) The curl ?=F(?.f_ sz *F;?
If F (x, y, z) is a vector function
i j k
curl£=VxE=i 3 3 :mt
— 6x ay 62
I?! F2 F3
= ii_31i.si[éa_§§s_ 2+ €52.21: ,;
6y 62 62 6x 6); ﬂy it is also known as rotation of F or called rot F. (d) The Laplacian operator is the divergence of the gradient of a scalar function. It is usually denoted by V2 _ ?_ﬂ 2 3: *3 e 3
dzv(gmd(g)=y_-Yg Vl"(‘ax+7aglk32)(‘ax+
6 c 6 A, 6 ,. 9‘ e‘ 9’
=V-iz+_g.1+.§. :9x’+9-’+az°
"' x 6y 62 5
1 ‘- 1‘ 'La, .
_ ag‘ 5g 5g”
_ (3762 + ayz 322 <9me «a Sc cum,
momentum“) vacrav
. 7‘
= Vzg (IS a scalar) My mommtwn
= Ag Laplacian
e.g. ‘1. classrcml ”MCI/amt:
. . A u 62
Electron kinetic energy operator 3? XE: 2P: Q.M transfer —-> -—2
2Me 2 6x
Mass 0/- a“
. A "712 n» 1
can be written as 36’ K5: 2V“
ZMe e
3n &_M, . 9
mamwrum optrdt'tcv (xmxc‘s) -—> if, = — PE 1 c“ sr‘mrﬂarﬁg fov g and z
r \ - "3.53. "721 ”>1
mama/41mm ‘ f, a “”30“” agrk 33)
=~mv a £3 9 .f. E 3
5;) ‘Z'i'l' S Maxwell’s e nations in electroma etism i D“ Ebi crn‘c vaohw (a)V __ __ =47?) Coulomb’ 5 law Z
(Electric ﬂux is proportional to charges inside)
Q
l magnetic Fiux
(b) _V_- E = 0 (Absence of free magnetic poles)
magnetic ﬂux is conserved
E .B.
1 BB , . .
(0) V x _1_)_—— — ~63; (Faraday slaw of Induction) moving charges creates magnetic ﬁeld ((1) Ex ><=B 4LJ+1§=E - Ampere’slaw " C +C a:
(force between current-carrying loops) ":2
D :5 E (electric permittivity) 0 O H = [.18 (magnetic permeability) K.E. 39=-1—mv2 =—1—ﬁ2
2 2m
)6=—z/‘1{—a—-+j—§-+123
x (By 62 J. Exact and inexact differentials :3) 11 TM rm 0 (Lu mm (‘c s" ,
internal a1 E = d7. lid“): (ctiﬁ‘arzncﬁt {5‘ exact)
en 0:99 2‘ _
s mre fun arm I" “a“ ‘0’ Hana rim? 3
(8X00? w‘ffzrmf‘fﬂi) fAESO ? (NS. (“e/aired 1‘0 m wfmpy OﬂdHFS WHOVKdG/IQ Am“ fd? .730 wipe/1:13 on Pam
Deﬁnition: If for any given differential d2: fi(xl!x2:x3:'”axnﬁxl +f2(xl’x2=”'xnﬂx2 + """ +fn,(x1’x2"”’xﬂ)dxn the changes in 2 do mt depend on the path in {x1 , 162,- » 'xn} —space, then
there must exist some function u = u(x,,x2,- - -x,,) such that dz is its total differential
g]: = fl(xux2v”' x11)
. , x
1.6. then necessarily
au
=fz(xi>xzv” xn)
6x2 If the above conditions are satisﬁed, then the quantity dz is an exact
differential. Euler’s Criterion for the exactness of a differential
This criterion is based-Ion the fact that if
= ﬁdx, + fzdxz + ...... then 3 a function u‘(x1,x2,x3,---) such that 23-1=ﬁ;% 2=f2;g 3=j.3; uuuuuu etc but since the value of higher-order second derivatives is n_o_t dependent on the order of
differentiation. alu 62a (72 677”, ~—-—-——-——'—'————.—_ 61516252 (”J‘Jtzaxl 6x, 5x2 or more generally if this is true for all different pairs of independent variable appearing in the differential,
then such a function u exists and may be determined. Example (i) Consider dz = (2172 ‘l' y)dz + (x " J’ —1)dy
\__._Y_J
Consider not equal!
dz is not an exact differential (with relationship of x and )2) (ii) dz = (2x + 3 y)dx + (3x — 4 y)dy %(2x+3y)=3 equal :> differential is exact 3(3x—4y)=3
8x , Since . ﬁc=—“=(2x+3y)
‘ 6x I2x+3ydx=x2 +3131 In themodynamics internal , ' d5 = DQ - DW (difference is exact)
energy \\
state fu/Ilction inexact differentials exact differentive ng=0 but CllDQ i 0. depends on path Tutorial Method of Least Squares b
Data points: (x1, yl), (x2, yz)....
' (Xi, Yi)
} a
x: independent variable 0
y: ' "‘a'epend'éhtvanabie One wishe to ﬁt the data with an equation of the form y = a + bx (1)
by determining the values of the coefﬁcients a and I: such that the discrepancy between the
values of our measurements yi and the corresponding values y = f(xi) given by Equation (1) is
minimized. One wants to extract from the data the most probable estimates for the coefﬁcients a and b. The problem is to establish criteria for minimizing the discrepancy and optimizing the
estimates of the coefﬁcients. For any arbitrary values of a and b, we can calculate the deviations Ay; between each of the
observed values yi and the corresponding calculated values. r Ayi =I yi—y = yi-a-bxi
If the coefﬁcients are well chosen, these derivations should be the smallest. Deﬁning X2 to be the discrepancy (total) X2 = )i2(Ayi )2 =2i1(yi — a — bxi )2 (sum of discrepancy squares) Least squares method -— obtain the values of a and b that can minimize X2
Using calculus: 6X2 _ 0x2 a — 0 and ———b— = O to obtain minimum values of a andb
a
6X2 6
=——— Z --a-bx. 2 =0
6a Vaa [i (y, J]
i: '2 23 (Yi‘a' Ext) 2 0
2.3 Yi =Na+b23xi (2) l H 6;: = 5% [2% (Vi 'a“bxi)2]= 0 = ‘2 E (Yi -a- bxi)xl = 0
2} xiyi = agari — bei2 (3) Solving equation (2) and (3) for the coefﬁcient a and b, this will give the values of the
coefﬁcient for which X2, the sum of squares of the derivation of the points from the calculated ﬁt, is a minimum. The solutions are: a = % ex? §yi -2i:xi gm) ' (4)
b $11an yea-axing ‘ <5)
K=N§xf -(§xi)2 . (6)
The quantities-need to be calculated are
N = 5i: 1 add up all the 1 (number of data points) Zilxi add up all the xi values §Yi add up all the y; values git? add up all the x? values Zilxi yi add up the product of my values. Look to your calculators. You can ﬁnd this summation buttons on the front panel. We do provide a computer program to perform this analysis in the laboratory. WW, Nifty (Mal flow To use Ladybird LS‘lSL (frt‘ - leaf: 7- Chin. 814,63, (0 ((796) Matrix Algebra
The manipulation of quantum mechanical equations and proofs and general _ relationships of quantum theory frequently require some rather involved algebraic and geometric operations. Most of these manipulations can be put into the compact form
by use of mathematical topic called MATRICES. A matrix is deﬁned as an ordered array of elements subject to certain rules of
operation. The elements may be real or complex are arranged in rows and columns: (9 A:
an 312 313 an:
2121
am1 3m Ais saidto beanmxnmatiix. ‘i l rowcolumn
Aisasquaremauixifm=n
Ais arowmatrix ifm= 1 &n>1
Ais acolumnmatrixifm>18cn=l Two matrix A & B are said to be equal if and only if
A=B=> aij=blj foralli,j A is a null matrix (i.e. represent by 0)
A=O ifajj=0 foralli,j A is the unit matrix
A = 1 if 213 = 55 where 6;]- is the Kronecker delta Unit matrices are square matrix, thus all diagonal elements are unity (1) and all , off-diagonal elements are 0. Unity matrix = cit-diagonal Diagonal The sum and difference of two matrices A & B (both In x n) is a new matrix C whose
elements Cij are the sum and difference of the corresponding elements of A & B. C=AiB => aij-Fbij=cij foralli,j
1. Matrix multiplication AB => (means) multiply the matrix B by the matrix A. (The order is important)
Matrix multiplication is not generally commutative. A is m x q & B is q x n The product C will be m x 11. (Two matrices A & B can be multiplied only if the left matrix A has the same
number of column as the right matrix B have rows.)
C has elements ij and the element ciJ- is obtained by taking the i111 row of A, placing it along-side. The 3' '1‘ column of B, multiplying each pair of elements and
adding all these together. The algebraic form is =2 aik bk}
where Jkthe summation index k adds the q, products arising from the‘ row by
column’ multiplications. AB at BA generally not commutative e.g.
&
h S ] [1z -1— 2g + 33 4+ 10 +1.3] 14. 32]
(1) 3 = 4+1G+1s r=+5=+a=~ = “a 77
1 U
(2) [a ii [a 1::[321 identity matrix
11. Matrices as Transformation Operator
The matrix equation
AX = Y we may regard A as an operator which transform the vector X into another vector 811 Hi2 an aln X1 31 1X1 + a12X2 €11an yl
an X2 = aztxi + azzxz aZIXn = Y2
anl am X11 3nle + ' 3:1an Ya A(Xl+X2)=AX1+AX2=:Y1+Y2 III. Determinants
Determinant of a matrix A, det A, which is associated with a square matrix A such as in
an 2112 '“' am
a m a
det A = 232‘ 2} 3“
anl an2 ... arm The elements of the matrix A are numbers or functions (real or complex). The
determinant is also a number or function whose value or form is determined by certain combinations of its elements.
If det A = 0, the matrix A is a singular matrix, otherwise non-singular. l x 1 matrix: the determinant is the element of itself. . Eaﬂ 312!
2 x 2 matrix: det A = 2121 a2: det A = 21113.22 —— 21122121 an 35:2 3113
321' 822 3123
an 892 3133 3x3mahixzdetA= . This determinant may be expanded as linear combination of 2 x 2 determinants
as follow I322 323' I311 313
dctA = an as: 33:; — an 331 53-3
&___.V___J
minor Ofa11= A11 32a 322!
331 33-2 + 3.13 det A = anAu — 2121 A12 + 6113 A13 detA = det A = 2111322343 + 3122173331 + 21132121332 -‘ 211221212133 -‘ 8112123332 ~ a13a22a31 if Special matrices and properties Matrix A with elements aij Transpose: AT = aJ-i [a is _ a c
e.g.2x2 A: a: d A=AT= It: cl Complex conjugate g a matrix A
(A33 = as; The adjoint of matrix A is deﬁned as its complex conjugate transpose is written as X"
(A411 = a‘ji' = (5%
A‘“ = A‘ A matrix is self-adjoint are said to be hermitian
i.e. A=A+=A* All hennitian matrices are square, it is also real and symmetric A square matrix is called orthogonal if its transpose is equal to its inverse .- A =A‘1 The inverse of A is A4
AA‘1 = ldetA 1 is the unity matrix
AA.“ = A‘IA = ldetA M . -* 4 mm It is customary to write complex numbers as z=x+iy Wherei= V—l
and x is the real part of Z and y is the imaginary part of Z
Re Z = x Im Z = y Complex numbers and complex functions occur frequently in quantum mechanics and
we have to become familiar with the properties and the algebraic manipulations
associate with complex numbers. Complex number can be represented as a point in an x—y Cartesian coordinate system
x = r cosG
y = r sine z = r (0039 + i sine) Complex plane imaginary axis real axis Two complex numbers are added by adding the real parts and the imaginary parts of
the two complex numbers
21 = X1 + iY1
Z2 = X2 + iyz
21i22=x1ix2+(y1iy2)i
The product of Z] and 22 is
2122 = (x1 + iy1)(x2 + iyz)
= we + new + iX1Y2+i2y1Y2
= (X1X2 ~ 3/1372) + i(X2Y1 + X1Y2)
e.g. 21 =—2+iandzz= l —Zi
21 + 22 = — 1 — i
z} — 22 = —3 + 31
m2 = (—2 +1)(1 —— 2i) = —2 + i + 4i —2i2
= 0 + 5i = Si (purely imaginary) An important quantity associated with any complex number 2 is its complex conjugate 2*. The complex conjugate of z is obtained by changing i to ~i 2: x +iy
z* = x — iy ‘
Z 2* = (X+iy)(x—iy) = x2 — 12y2 =‘ x2 + y2 i2 = -1, i3 = —i and i4 = 1 I Z l = (225% = (X2 + y2)'/2 afz:z*, MY x+£9=x“‘9" 255:0 $920- 2 "am; only read fmf (in-3' lf’iS'f‘r‘TcﬁGmiC’" XréVC‘L ’70 9 firm) ,
£3
£5— CEEﬁﬁi364Sl Dirac Notation When we work with the various functiohs it is often
an advantage to use a shorthand notation developed by Diraco
This is an extension of the < > notation for expectation values:— mm is written |m>, called ket
* l l wn is written <nl5 called bra f_m'wn mm dT <nim> flea ‘l’n Q 1pm dT <n1le> As a simple example of the use of this notation; start with A
; Hln> = E |n> limp]?! = End) 11 I1 Premultiply by w; and integrate: I°° J” :3: d ~ a f°°' J“ 6. ~—
—w wn I‘Dn T * ﬁn ~w 1pn LJJn- T " En 3
<n|Hln> = En<nln> = En 5mm 6mm is the Kronecker delta symbol.
(if n = m) 1
anal : { -
(if n # m) 0 ...

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