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Title: Mathematical Physics-I Mock Papers
Description: Compilation of semester exam level questions in Mathematical Physics-I on the following topics. LINEAR ALGEBRA - linear vector spaces, metric and inner product spaces, function spaces, span and basis, orthonormal basis and transformation, Gram-Schmidt procedure, Schwarz inequality, Bessel inequality, linear operators, identity and inverse, commutators, adjoint, hermitian, skew-hermitian, unitary, and orthogonal operators, basis expansion, matrix representation of functions and operators, operator transformation, eigenvalues and eigenfunctions, diagonalisation, spectral decomposition, expectation values, simultaneous eigenstates and commutativity, normal matrices, super-operators. COMPLEX ANALYSIS - complex functions of complex variables, multivaluedness, continuity, differentiability and derivative, Cauchy-Riemann conditions, singularities, analytic and entire functions, contours, simply connected regions, Cauchy integral theorem, multiply connected regions, barriers, Cauchy integral formula and derivatives, Morera’s theorem, Cauchy’s inequality, Liouville’s theorem, neighbourhood and radius of convergence, Taylor and Laurent series, poles, simple poles, and essential singularities, meromorphic functions, branch points, analytic continuation, residue theorem, Cauchy principal value, Mittag-Leffler expansion, counting poles and zeroes, Rouche’s theorem, complex definite integrals, branch cuts and periodicity, evaluation of sums.
Description: Compilation of semester exam level questions in Mathematical Physics-I on the following topics. LINEAR ALGEBRA - linear vector spaces, metric and inner product spaces, function spaces, span and basis, orthonormal basis and transformation, Gram-Schmidt procedure, Schwarz inequality, Bessel inequality, linear operators, identity and inverse, commutators, adjoint, hermitian, skew-hermitian, unitary, and orthogonal operators, basis expansion, matrix representation of functions and operators, operator transformation, eigenvalues and eigenfunctions, diagonalisation, spectral decomposition, expectation values, simultaneous eigenstates and commutativity, normal matrices, super-operators. COMPLEX ANALYSIS - complex functions of complex variables, multivaluedness, continuity, differentiability and derivative, Cauchy-Riemann conditions, singularities, analytic and entire functions, contours, simply connected regions, Cauchy integral theorem, multiply connected regions, barriers, Cauchy integral formula and derivatives, Morera’s theorem, Cauchy’s inequality, Liouville’s theorem, neighbourhood and radius of convergence, Taylor and Laurent series, poles, simple poles, and essential singularities, meromorphic functions, branch points, analytic continuation, residue theorem, Cauchy principal value, Mittag-Leffler expansion, counting poles and zeroes, Rouche’s theorem, complex definite integrals, branch cuts and periodicity, evaluation of sums.
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3
Qucstion 1 The set of functions Po = 1, P = I, 2 = * ; are orthogonal
with nit weight on the range -1 < < +I
...
a
...
3
...
Find the unitary matrix Uthat transforms from thc normalized P, basis to the nor
Imalized E basis
...
Find the nitary matrix V that trans•orns fromthe normalizcd E, basis to the nor
malizcd P basis
...
Expand f(r) =5r? -3r + I in terns of the nomnalizcd versions of both bases, and
verify tlhat the transformnation matrix U coverts the P-basis cxpansion of f() into its
F-basis cxpansion
...
Jo) = Cye-r, Jos) = Czc-r}}spans
spans aa threc
space ol functions
...
a
...
(3)
b
...
Writc the cxplicit form of M' which is thc matrix M writton in a dilcrent ON basis,
B = {lo)lo,)lo3)} using the unitary givcn below (The matrix elements of U are
Uij i= (0,, o;)) (2):
0
æ=
1//2 -i/V2
0 1/V2 i/2
(1)
d
...
rt
...
3|
c
...
r
...
lo),)
...
(3
Qucstion 3 A, B
...
The ON cigonvectors of Aare u)u2)
...
a
...
n
...
Consider tlhe natrix
|D,C1
...
Lct
...
Allother cigenval1es of A and B arc less thap 1/2
...
Specify the range over which this
expansion holds
...
Specify the range over which this expansion holds
...
For this you may need to use the result that
() an-s
n=
Question 2 (a) Show that all roots of f(z) = z 4
than 10
...
1
[6]
cot z
z-ne
=n-oo
Question 3Evaluate the integral
(1)
You may need to use
tan-1
de
(2)
for aand bpositive with ab < 1, by choosing appropriate contours and checking that
the integrand satisfies the necessary conditions on different parts of the contour
...
10|
Question 5 (a) The matrices, La, Ly, Lz, representing angular momentum
components are all Hermitian
...
5]
(b) Expand the function e* in terms of the Laguerre polynomials L() which are
orthonormal on the range 0 S a <0 with scalar product (flg) = f(e)g(z)e-*dr,
keeping only the first four terms in the expansion
...
[5)
T2)= UtIV
Cauchy- Riemann Conditon
fel: analy tic, sinply cowmected veglon Carchy Integval Theoven
fe) de=0, Ciany chsed contour Within vegin af analy tity
foy any CyCz
f2) dz jG :continusUs defovmatiom
6 fzd2>
fovmula
val
Integ
o
Cavchy
i
i
C
withim
z
f(z):
t2) d2e f(2
...
:sing ulav point)
Zo'egulav point
fz) =
n
flz)=
L6)dz
2-2)"
Lavrent Erpansidn
ovder n at z=z
of
pole
exists;
liit
ny
smallest
s
flz);
f aroundz
(e-zy
of
Lt
LE
(t
in
a,
Residve:
vesidves)
of
Within C
6fle)dz= 2ri (sUm
Lt
Resi dve Theovem
((z-2)
Contour
com
on
larity
3:singu simple pole)
Sfa) dz (aenevaly
Lt
t
=
t
=
)
d
z
ffo) dz
a-ijother then 2
...
Co
fz)
lal
N¹
62dz = 2i(N¡-P+) Pe: no
Title: Mathematical Physics-I Mock Papers
Description: Compilation of semester exam level questions in Mathematical Physics-I on the following topics. LINEAR ALGEBRA - linear vector spaces, metric and inner product spaces, function spaces, span and basis, orthonormal basis and transformation, Gram-Schmidt procedure, Schwarz inequality, Bessel inequality, linear operators, identity and inverse, commutators, adjoint, hermitian, skew-hermitian, unitary, and orthogonal operators, basis expansion, matrix representation of functions and operators, operator transformation, eigenvalues and eigenfunctions, diagonalisation, spectral decomposition, expectation values, simultaneous eigenstates and commutativity, normal matrices, super-operators. COMPLEX ANALYSIS - complex functions of complex variables, multivaluedness, continuity, differentiability and derivative, Cauchy-Riemann conditions, singularities, analytic and entire functions, contours, simply connected regions, Cauchy integral theorem, multiply connected regions, barriers, Cauchy integral formula and derivatives, Morera’s theorem, Cauchy’s inequality, Liouville’s theorem, neighbourhood and radius of convergence, Taylor and Laurent series, poles, simple poles, and essential singularities, meromorphic functions, branch points, analytic continuation, residue theorem, Cauchy principal value, Mittag-Leffler expansion, counting poles and zeroes, Rouche’s theorem, complex definite integrals, branch cuts and periodicity, evaluation of sums.
Description: Compilation of semester exam level questions in Mathematical Physics-I on the following topics. LINEAR ALGEBRA - linear vector spaces, metric and inner product spaces, function spaces, span and basis, orthonormal basis and transformation, Gram-Schmidt procedure, Schwarz inequality, Bessel inequality, linear operators, identity and inverse, commutators, adjoint, hermitian, skew-hermitian, unitary, and orthogonal operators, basis expansion, matrix representation of functions and operators, operator transformation, eigenvalues and eigenfunctions, diagonalisation, spectral decomposition, expectation values, simultaneous eigenstates and commutativity, normal matrices, super-operators. COMPLEX ANALYSIS - complex functions of complex variables, multivaluedness, continuity, differentiability and derivative, Cauchy-Riemann conditions, singularities, analytic and entire functions, contours, simply connected regions, Cauchy integral theorem, multiply connected regions, barriers, Cauchy integral formula and derivatives, Morera’s theorem, Cauchy’s inequality, Liouville’s theorem, neighbourhood and radius of convergence, Taylor and Laurent series, poles, simple poles, and essential singularities, meromorphic functions, branch points, analytic continuation, residue theorem, Cauchy principal value, Mittag-Leffler expansion, counting poles and zeroes, Rouche’s theorem, complex definite integrals, branch cuts and periodicity, evaluation of sums.