Mathematical Physics Sample Exam 1 Submit fully explained solutions to the following: 1. Give one example example each each of an elliptic, elliptic, hypebo hypebolic, lic, and paabolic paabolic patial patial diffe diffeentia entiall e!uation. e!uation. [1 pt] ". #ompletely #ompletely wo$ wo$ out the diffee diffeential ntial e!uation e!uationss that esult esult fom an applicatio application n of the method method of sepaation of vaiables to the %elmholt& e!uation in #icula'#ylindical, and Spheical #oodinates. ()f$en has an indication of what the esults should be, you ae expected to suppl y the details* [4 pts] 2
2
∂ u ∂ u 1 ∂ u + α = 2 2 ,t , t > 0 ,− L < x < L 2 ∂ t v ∂ x ∂ t
+. Solve Solve the the dam dampe ped d wave wave e!ua e!uatio tion n
with the bounday
conditions:
u (− L , t ) =u ( L L , t )=0, t ≥ 0 , πx u ( x x , 0 )= sin , − L≤ L ≤ x≤ x ≤ L , L
( )
|
∂ u ( x , t ) ∂ t
=1, − L≤ L ≤ x≤ L
. [3 pts]
t = 0 2
. #alcul #alculate ate the the GeenGeen-ss functi function on Gx,xGx,x-/. /. 0ote 0ote
d ' ' G ( x x , x ) + G ( x x , x )= δ ( x − x ' ) with the 2 dx
' bounday conditions G ( − L , x' ) =G ( L L , x ) = 0 using two methods: ). Splitting the egion (',* (',* into one egion whee x2x- and anothe egion egion x3x- and imposing the appopiate bounday conditions. [2 pts] 4. 5sing an othogon othogonal al function function expansion expansion involving involving solutions solutions of the eigenvalue eigenvalue poblem poblem 2
d φn+ λn φn= 0 6igue out what the appopiate bounday conditions ae/ [2 pts] 2 dx 2
7. Solve
d y x ( x ) + y ( x x )= sin ( x ) with the bounday conditions y 2 dx
8. 9s the the linea linea diff diffee eenti ntial al ope opeato ato
(
2
d −2 x d + 2 n 2 dx dx
)
L ) =0 (− L ) = y ( L
. [3 pts]
self'adoint; 9f it is not, what integating
facto is needed to ma$e it self'adoint; [2 pts]
<. 5se the Gam'Schmid Gam'Schmidtt othogonali&a othogonali&ation tion scheme scheme to geneate the fist fist thee othono othonomal mal functions functions stating with the following: n 9nitial non'othogonal basis: un ( x x ) = x , n =0,1,2,3 … =eight function: 9nteval:
w ( x x )= 1 / √ 1− x
−1 < x < 1
2
[2 pts]
>. 5se the least least s!uaes citeio citeion n to detemine detemine the best appox appoximatio imation n to sinx/ as as a linea function function $x on on the inteval 0 < x < A whee ) is some positive numbe. [1 pts]
Mathematical Physics Sample Exam " 2
∂u ∂ u = ,t > 0 ,− L < x < L 1. Solve the e!uation ∂ t ∂ x 2 u ( − L , t ) = u ( L , t )=0, t ≥ 0 , πx u ( x , 0 )= sin , − L≤ x≤ L . [20 pts] L
with the bounday conditions:
( )
2
". )/ #alculate the Geen-s function Gx,x-/. 0ote
G ( − L , x ) =G ( L , x ) = 0 '
the bounday conditions
'
d d ' ' G ( x , x ) + G ( x , x )= δ ( x − x ' ) with 2 dx dx [10 pts]
2
4/ Solve
d d 2 2 y ( x ) + y ( x ) = L − x with the bounday conditions y (− L ) = y ( L ) =0 . 2 dx dx
[10pts]
+. 5se the Gam'Schmidt othogonali&ation scheme to geneate the fist thee othonomal n functions stating with the initial non'othogonal basis: un ( x ) = x , n = 0,1,2,3 … =eight function: w ( x ) =1 ? and 9nteval: @ ¿ x < 1 [20 pts] . )/ #alculate the +A Geen-s function G ( r , r ' ) whee ⃗
G( r ,r ⃗
⃗
)
'
⃗
∇
2
' ' G ( r , r ) =δ ( r − r ) and ⃗
⃗
appoaches @ at infinity. [10 pts]
4/ Solve fo
Φ( r ) : ⃗
∇
2
Φ ( r )= ⃗
{
1
, r ≥ R
N
r
1
, r ≤ R
N
R
where N > 0 . [10 pts]
⃗
⃗
Mathematical Physics Sample Exam + Submit fully explained solutions to the following: ∞
1. Evaluate
cos ( nx ) , x ∈ ( 0,2 π ) ∑ n = 2
. [2 pts]
n 1
". Bepesent f ( x ) = x 2 ? @ 2 x 2 ", as: a/ a full'ange 6ouie seies, b/ a half' ange sine seies. c/ a half'ange cosine seies. [3 pts] +. #alculate the 6ouie tansfom of
f ( t )=cos ( t ) exp (−! t + "t ) # ! , " , > 0 2
2
. Solve fo
u ( x , y ) # 0 < x < α , 0 < y < $ :
[2 pts]
2
∂ u ∂ u + 4 2=0 2 ∂x ∂y
u ( x , 0 )= u ( x , $ ) =0, 0 < x < α u ( 0, y ) = u ( α , y ) =% , 0 < y < $ [3 pts] 2
7. Solve fo yx,t/:
2
∂ y ∂ y ∂ y = 2 + # y ( x , 0 )= f ( x ) # ∂ y ( x , t =0 ) =& ( x ) . [3 pts] 2 ∂ t ∂ t ∂ x ∂ x
8. Aetemine the invese aplace tansfom of
1 + + 3
−
2
by using a/ patial factions and b/
4omwich invesion. [3 pts] 3
<. Solve fo xt/:
d x + x = δ ( t −( ) + sin ( t ) # , ( > 0 . [4 pts] 3 d t
Mathematical Physics Sample Exam ∞
1. Evaluate
sin ( nx ) , x ∈( 0,2 π ) ∑ n = 2
. [20 pts]
n 1
2
". Solve fo y ( x ,t ) :
2
∂ y ∂ y ∂y = 2 , y ( x ,t = 0 )=e− x , ( x , t = 0 )=0 . [20 pts] 2 ∂t ∂ t ∂ x 2
+. ). Evaluate the invese aplace tansfom of
1
' √ ' ( ' + 9 )
x
2
4.Evaluate the aplace tansfom of ) ( x )=∫ dt ' ' e
t ''
. Solve fo x 1( t ) and 2
x 2( t )
[20 pts]: 2
d x 2 = − + =+ ( x 1− x 2 ) * + x x ) sin t * ? ( ) ( ) 2 1 1 1 2 2 d t d t 0 ∧d x1 dx ( t = 0 )=0 # 2 ( t =0 )=0 whee x 1 ( 0 ) = x2 ( 0 ) = dt dt d x 1
t '
∫ d t ' e ∫ dt e−
−3 t ' '
0
. (10 pts] − 2t '
0
t
0
[10 pts]
Mathematical Physics Sample Exam 7 x
1. Solve fo φ ( x ) : φ ( x )= x −∫ ( t − x ) φ ( t ) dt . [20 pts] 0
∞
".Solve fo φ ( x ) : x =∫ e 2
2
−( x −t )
φ ( t ) dt . [20 pts]
−∞
∞
%ints:
n
t − t + 2tx n ( x ) =e ,−∞ < x < ∞ , n n= 0
∑
2
+∞
∫
−∞
− x 2
e n ( x ) dx =2 n- √ π n
1
2 +.Solve fo the eigenvalues and eigenfunctions: φ ( x )= λ ∫ ( 2 t − x ) φ ( t ) dt [20 pts] 0
.Che pobability density of momentum of a one dimensional gas consisting of molecules with mass m is f . ( / )= N e−α ( /− / ) whee α ∧ /0 ae given positive constants and 0 is a nomali&ation constant that needs to be detemined. #alculate the following: a/ 0omali&ation constant 0 [5 pts] b/ mean momentum [5 pts] c/ vaiance of the momentum [5 pts] 2
0
2
d/ pobability density of the $inetic enegy
/ f 0 ( 0 ) whee 0 = 2*
[10 pts]
7.) andom wal$e stats fom the oigin and ta$es steps of length 1 eithe going to the left 1 x =−1 /o going to the ight left 1 x =+ 1 / . Che pobability of ta$ing a step to the ight o left on the 1st step, +d step, 7th step, etc ae D and espectively. Che pobability of ta$ing steps to the ight o left on the "nd step , th step, 8th step, etc ae F and F espectively. #alculate the following: a/ Pobability that the andom wal$e is at the position x@ afte six steps. [5 pts] b/ Expectation value of the position of the wal$e afte "0 steps. [5 pts] c/ Second moment of the position of the wal$e afte "0 steps. [5 pts] 8.6o a adioactive sample, 1@ decays ae counted on aveage in 1@@ seconds. Estimate the pobability of counting 7 decays in 1@ seconds. [5 pts]