# (1) (a) To prove the Wien`s Law :

```(1) Black-body radiation
To prove the Wien's Law :
max T  constant

d ( )
d
let
x
6
h c /  kT
 1) 1  5 (e h c /  kT  1)  2  e h c /  kT 
  max   5 (e
 max kT
hc
1
x
hc2
kT
   max
0
we have :
1
x
 5 x(e  1)  e  0
by u sin g Newton' s method, we obtain :
x  0.2014052  max T  0.00289779mK
(2) The Uncertainty principle (melons in quantum land)
(a) Why do we have to be careful when cutting open melons grown in quantum land ?
  x   Px 
 V x 
h
4
h
, m  0.0001kg,  x  0.2m,
4 m x
V x  25 m / s  Very fast !
The seeds leave the melon with this velocity when it is cut open. So you should be
careful.
(b) Are such melons visible?
No, Let's consider the recoil of a melon at reflection of a &quot;visual&quot; photon. Because the
peel of the melons is very hard, the collision can be treated as elastic collision, then
the momentum of the melon after the collision is approximately :
PM 2 Pphoton 
h

 10 4 kg m / s
 v  5 km / s  very fast
By the time this photon is seen, the melon is already situated elsewhere
```