1000 Solved Problems in Modern Physics

(Tina Meador) #1

112 2 Quantum Mechanics – I


2.37 (a) By definition the magnetic moment of electron is given by the product of
the charge and the areaAcontained by the circular orbit.


μ=iA=−

eπr^2
T

=−

ωeπr^2
2 π
=

−emeωr^2
2 me

=−

eL
2 me

(b)μl=−
e
2 mc

(L(L+1))^1 /^2 

μs=−(2e/ 2 mc)(S(S+1))^1 /^2 

2.38 The principle of the Stern–Gerlah experiment is described in Problem 2.35.
While the atom is under the influence of inhomogeneous magnetic field the
constant force acting on the atom along y-direction perpendicular to the
straight line path OAF in the absence of the field, is a parabola (just like
an object thrown horizontally in a gravitational field). The equation to the
parabola is
y=kx^2 (1)
wherekis a constant, Fig 2.3. Let us focus on the atom which deviates upward.
After leaving the field atD, its path alongDEis a staright line. It hits the
plate atEso thatEF=s. WhenEDis extrapolated back, let it cut the line
OA FinC.
Taking the origin atO, Eq. (1) satisfies the relation atD,
h=kl^2 (2)
Furthermore atD,
(
dy
dx


)

D

= 2 Kx|D= 2 K.OA= 2 K.l (3)
(
Dy
dx

)

D

=

AD

CA

=

h
CA

(4)

Combining (2), (3) and (4), we get

CA=

l
2

(5)

Now the time taken for the atom along thex-component is the same as for
along the y-component. Therefore

t=

l
ν

=

(

2 h
a

) (^12)
(6)
or
h=
l^2 a
2 ν^2


(7)
Free download pdf