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A. QUANTUM PHYSICS
Question 1:
The idea introduced by Max Planck to precipitate the creation of quantum physics is that the
energy and the momentum can be quantized so as to deduce the formula for the observations
of the frequency relied on the energy generated and emitted by any black body, and this idea is
referred to Planck's law which consists of the Boltzmann distribution and is also applicable in the
classical physics.
The experimental motion behind this introduction of the concept was that Planck wanted to find
the spectral density of electromagnetic radiation generated and then emitted by the black body.
That happened especially when he turned his attention to the issue of radiation from black-body
objects, as he questioned himself about the relationship between the intensity of the
electromagnetic radiation and the radiation frequency, but he found no theory agree with any
experimental results during that time.
Question 2:
(a) The uncertainty principle can be described as the product of the change of the position and
the change of the momentum is always greater or equal to a half of the Planck constant which is

= 6.626 10−34 m2  kg / s . In other words, it can be said from the uncertainty principle that
the more accuracy we know about the position implies the less accuracy we know about the
momentum, and its converse is true.
The uncertainty principle is indeed the fundamental limit rather than the limit of our
experimental apparatus. And that is because the formula for the uncertainty principle is

x p 

2

, and there is no way we can do with our experimental apparatus to change x

without changing  p in the opposite direction and vice versa.
(b) Let d nucleus be the diameter of the nucleus of the atom, which usually ranges from

1.6  10 −15 m (for the lightest atom) to 15  10 −15 m (for the heaviest atom). In other words, we
have 1.6 10−15 m  d nucleus  15 10−15 m .
According to the principle of uncertainty, we have:

x   p 

2

Since it is given that the proton is confined to the nucleus, it follows that:

x  d nucleus  15  10−15 m
Since  p is the momentum of the proton, we have p = m  v where m is the mass of the
proton which is 1.67 10−27 kg , and v is the uncertainty in its speed that we need to find.

Hence, the uncertainty in the speed of the proton can be found by:

x  p 

2

 x  mv 
 v 
 v 

2mx

2


2md nucleus



2m (15 10−15 )

1.05  10−34 J .s
2 (1.67 10−27 kg )(15 10−15 m )

 v  2.10  106 m / s
Hence, the uncertainty in speed was found to be 2.10  106 m / s , as desired result.
Question 3:




We consider the following normalized state  which satisfies P  +; S z = +


 = 20% and
2



P  −; S z = −  = 80% . In fact, we denote the following:
2


 a + bi 

 =

 c + di 
where a, b, c and d are the real-valued number. Therefore,

 =  a − bi c − di 
Since  is in its normalized state, we must have  | = 1 . Hence, it follows that:

 | = 1
 a + bi 
  a − bi c − di  
 =1
 c + di 
 ( a − bi )( a + bi ) + ( c − di )( c + di ) = 1
 a 2 + b2 + c2 + d 2 = 1
Since 20% of the atoms have spin-up along the z-axis, we have:



P  +; S z = +  = 20%
2

 + z |

2

= 0.2
2

 a + bi 
 1 0 
 = 0.2
 c + di 
 a + bi = 0.2
2

 a 2 + b 2 = 0.2
Since 80% of the atoms have spin-up along the z-axis, we have:



P  −; S z = −  = 80%
2

 − z |

2

= 0.8
2

 a + bi 
  0 1 
 = 0.8
 c + di 
 c + di = 0.8
2

 c 2 + d 2 = 0.8
Hence, the percentage of atoms measured spin-up configuration along the x-axis or S h = +
calculated by:



P  +; S x = +  = + x | 
2


  1
 P  +; S x = +  = 
2  2


2

1   a + bi 


2   c + di 

2

1
1


 P  +; S x = +  =
( a + c ) +...