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
OAFinC.
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 getCA=
l
2(5)
Now the time taken for the atom along thex-component is the same as for
along the y-component. Thereforet=l
ν=
(
2 h
a) (^12)
(6)
or
h=
l^2 a
2 ν^2