Physics 550 Problem Set ff2 Due Mon. Sept. 23'h Note: o Do not write up a solution that is...

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Physics 550 Problem Set ff2 Due Mon. Sept. 23'h Note: o Do not write up a solution that is essentially no more than a string of equations. Use language to show that you know what the meaning of your solution ii. When grading I am looking for understanding and will not consider strings of equations as indicating understanding. o Avoid the temptation of trying to do the entire assignment the night before it is due. This one is impossible to do that way. -You learn much less that way and it is rarely helpful for you. 1.
	
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550
Problem Set 
Due Mon. 
ff2 Sept. 23'h
Note:
o Do not 
write up a solution that 
is essentially no more 
than a string of 
equations.
Use language to show that you 
know what the meaning your 
of solution 
ii. When
grading 
I am looking 
for understanding and 
will not consider strings 
of equations
as indicating 
understanding.
o 
Avoid 
the temptation of trying 
to do the entire assignment 
the night before it 
is
due. This one is impossible 
to do that way. learn 
-You much less that way 
and it
is rarely helpful for you.
1. Find the order of the group 
of transformations 
that leaves an 
equilateral triangle
invariant. Identify 
the complete set of group 
elements, find a matrix 
representation, and
then construct the 
multiplication table.' 
What is the common 
name of this group?
2. Show that the two expressions 
below are equivalent.JrV
: 
(p,) 
Oo.- e.(p,)p*e(p,)
(1)
I* 
: 
(p,) 
o* v.@1Pv@)
(2)
l* 
where
: 
o(r,l 
o* 
*r,l
(3)
"i*o(a\) 
hf 
What conditions are necessary 
for this equivalence?.
-;t5s")'#\
,1
the position-momentum 
perive Heisenberg 
Uncertainty 
Principle. 
What is the value
*tr3) 
i' 
"- 
of L,rA,p* 
for the Gaussian wave packet
\rf 
,t 
@):@fur""el-@#A+f,no@-,,)] 
e)
Suppose the variance, 
02 is the complex 
number o2o lol, 
+ then what 
is the value of
A,rLp* (be careful 
about normalization)? 
What is thL 
soluiion to 
the Schrodinger 
equa-
tion of a free particle 
it tlt(r) is 
taken as the initial 
condition ,b(*,t:0)?
4. For the wave function, : 
,h@) csexp(i,psrlfr)lr/AT$, 
find the normalization 
con_
stant c6 and the probability 
of finding the particle 
in the 
interval +"ltfg. 
what is the
mean momentum?

David · Accepted Answer

Answer 
In an equilateral triangle we know that all sides are equal and all angles are of 60
0
.  The set of 
operation which will leave the triangle invariant will be, Under reflections in the three planes 
through the vertices The above two operations will form a group of six operations which needs to 
be performed to make in invariant. This kind of group elements are known as dihedral group and 
represented by D3. 
Rotations about z – axis both clockwise and anticlockwise we get,
and on counting reflections in the all three planes;
So the set of operator is called D3 = {e, a, a
2
, b, c, d}. 
Here we will count the anticlockwise rotations and positive while clockwise rotations and 
inverse rotations so that the inverse of e, a, a
2 
are 
e
-1
 = e, a
-1
 = a
2
, and a
-2
 = a. 
So, The inverses of the reflection operators are the operators themselves: 
b
-1
 = b, c
-1
 = c, and d
-1
 = d 
Since the inverse operators are coming back to the same vales hence the operators will leave 
equilateral triangle invariant.

Physics 550 Problem Set ff2 Due Mon. Sept. 23'h Note: o Do not write up a solution that is essentially no more than a string of equations. Use language to show that you know what the meaning of your...

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