26-10-2023

1. 1
Consider a mass–spring system
where a 4 kg mass is attached to a
massless spring of constant
k = 196 Nm-1
; the system is set to
oscillate on a frictionless, horizontal
table. The mass is pulled 25 cm
away from the equilibrium position
and then released.
(a) Use classical mechanics to find
the total energy and frequency of
oscillations of the system.
(b) Treating the oscillator with
quantum theory, find the energy
spacing between two consecutive
energy levels and the total number
of quanta involved. Are the
quantum effects important in this
system?

2. 2
Solution

3. 3
2
1
( )
According to classical
mechanics
1
=
2
1 196
2 3.14 4
1 49 .
6.28 . .
1
7
6.28
1.115
a
k
m
N
X kgXm
kg m
kg m s
X Xs
Hz







2
2
2
Also,
1
2
1
196 (25 )
2
98 (0.25 )
98 0.0625
6.125
E kA
N
X X cm
m
N
X m
m
NX m
J






4. 4
( )
According to Planck's postulate,
the energy of an oscillator of
natural frequency must be an
integral multiple of , i.e.
, 0,1,2,3,....(1)
where is a universal constant &
is the energy
b
h
E nh n
h
h




 
of a "quantum"
of radiation.
34
34
34
34
34
33
Using the values of , &
in eq.(1), we get
6.125J 6.626
10 . 1.115
6.125 7.39
10 .
6.125 7.39 10
6.125 10
7.39
0.83 10
8.3 10
E h
nX X
J sX Hz
J nX
X J sXHz
nX X
X
n
n X
X





 
 
 
 


5. 5
34
34
No, the quantum effects are
not important in this system
because the energy of one
quantum, i.e.
6.626X10 J.s
X1.115Hz 7.39X10 J in
comparison with total energy
6.125 J is so small that it may
be neg
h 



lected and the number
of photons is very large.