Angular Momenta
Q36(1) The spherical harmonics Y 1,1(θ, φ), Y 1,0(θ, φ), Y 1,−1(θ, φ) are given by Eqs. (16.65) to (16.67). In Cartesian coordinates, these functions are given by 1 Y 1 , − 1 ( x , y , z ) = 3 8 π x − i y r , Y 1 , 0 ( x , y , z ) = 3 4 π z r , Y 1 , 1 ( x , y , z ) = − 3 8 π x + i y r . (a) Using t...
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| description | Q36(1) The spherical harmonics Y
1,1(θ, φ), Y
1,0(θ, φ), Y
1,−1(θ, φ) are given by Eqs. (16.65) to (16.67). In Cartesian coordinates, these functions are given by
1
Y
1
,
−
1
(
x
,
y
,
z
)
=
3
8
π
x
−
i
y
r
,
Y
1
,
0
(
x
,
y
,
z
)
=
3
4
π
z
r
,
Y
1
,
1
(
x
,
y
,
z
)
=
−
3
8
π
x
+
i
y
r
.
(a) Using the expression in Cartesian coordinates for the quantised orbital angular momentum operator
L
^
c
z
in Eq. (27.86), verify by explicit calculations that Y
1,1(x, y, z), Y
1,0(x, y, z)and Y
1,-1(x, y, z) are the eigenfunctions of
L
^
c
z
corresponding to eigenvalues ħ, 0 and –ħ.
(b) Consider the following coordinate transformations:
x
→
z
′
,
y
→
x
′
,
z
→
y
′
.
(1)
202Show that the component of the orbital angular momentum operator along the z′-direction is the same as that along the x-direction, i.e., show that
L
^
c
z
′
=
L
^
c
z
.
(2) Show that the simultaneous eigenfunctions of
L
^
c
2
and
L
^
c
x
corresponding to eigenvalues of
L
^
c
2
equal to 2ħ
2are given in Cartesian coordinates by
X
1
,
−
1
(
x
,
y
,
z
)
=
3
8
π
y
−
i
z
r
,
X
1
,
0
(
x
,
y
,
z
)
=
3
4
π
x
r
,
X
1
,
1
(
x
,
y
,
z
)
=
−
3
8
π
y
+
i
z
r
.
SQ36(1)(a) In Cartesian coordinates we have
2
:
p
^
x
Y
→
1
,
−
1
:
=
−
i
ℏ
3
8
π
∂
∂
x
x
−
i
y
r
=
−
i
ℏ
3
8
π
(
1
r
−
x
−
i
y
r
2
∂
∂
x
r
)
=
−
i
ℏ
3
8
π
(
1
r
−
x
−
i
y
r
2
x
r
)
=
−
i
ℏ
3
8
π
1
r
3
(
r
2
−
x
2
+
i
x
y
)
.
p
^
y
Y
→
1
,
−
1
:
=
−
i
ℏ
3
8
π
∂
∂
y
x
−
i
y
r
=
−
i
ℏ
3
8
π
(
−
i
r
−
x
−
i
y
r
2
∂
∂
y
r
)
=
−
i
ℏ
3
8
π
(
−
i
r
−
x
−
i
y
r
2
y
r
)
=
−
i
ℏ
3
8
π
1
r
3
(
−
i
r
2
−
x
y
+
i
y
2
)
.
L
^
z
Y
→
1
,
−
1
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
−
1
=
−
i
ℏ
3
8
π
1
r
3
(
(
−
i
x
r
2
−
x
2
y
+
i
x
y
2
)
−
(
y
r
2
−
y
x
2
+
i
x
y
2
)
)
=
−
i
ℏ
3
8
π
1
r
3
(
−
i
x
r
2
−
y
r
2
)
=
−
ℏ
3
8
π
1
r
{
x
−
i
y
}
=
−
ℏ
Y
1
,
−
1
.
203Next we have
p
^
x
Y
→
1
,
0
:
=
−
i
ℏ
3
4
π
∂
∂
x
z
r
=
−
i
ℏ
3
4
π
(
−
z
r
2
∂
∂
x
r
)
=
i
ℏ
3
4
π
z
x
r
3
.
p
^
y
Y
→
1
,
0
:
=
−
i
ℏ
3
4
π
∂
∂
y
z
r
=
−
i
ℏ
3
4
π
(
−
z
r
2
∂
∂
y
r
)
=
i
ℏ
3
4
π
z
y
r
3
.
L
^
z
Y
→
1
,
0
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
0
=
x
(
i
ℏ
3
4
π
z
y
r
3
)
−
y
(
i
ℏ
3
4
π
z
x
r
3
)
=
0.
Finally we have
p
^
x
Y
→
1
,
1
:
=
i
ℏ
3
8
π
∂
∂
x
x
+
i
y
r
=
−
i
ℏ
3
8
π
(
1
r
−
x
+
i
y
r
2
∂
∂
x
r
)
=
i
ℏ
3
8
π
(
1
r
−
x
+
i
y
r
2
x
r
)
=
i
ℏ
3
8
π
1
r
3
(
r
2
−
x
2
−
i
x
y
)
,
p
^
y
Y
→
1
,
1
:
=
i
ℏ
3
8
π
∂
∂
y
x
+
i
y
r
=
i
ℏ
3
8
π
(
i
r
−
x
+
i
y
r
2
∂
∂
y
r
)
=
i
ℏ
3
8
π
(
i
r
−
x
+
i
y
r
2
y
r
)
=
i
ℏ
3
8
π
1
r
3
(
i
r
2
−
x
y
−
i
y
2
)
,
L
^
z
Y
→
1
,
1
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
1
=
i
ℏ
3
8
π |
| doi_str_mv | 10.1201/9780429296475-36 |
| format | Book Chapter |
| fullrecord | <record><control><sourceid>proquest_infor</sourceid><recordid>TN_cdi_proquest_ebookcentralchapters_6320223_43_214</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>EBC6320223_43_214</sourcerecordid><originalsourceid>FETCH-LOGICAL-i1094-9cc9fa2dddc04a250877a294638d6536fd99312b453a49138476e61547ef44013</originalsourceid><addsrcrecordid>eNpVkMtOAzEMRYMQCCgVW5b8wEAcOw8vqwooEogNrKN0HjAwnZRkCuLvmVI2eGPZ8j26vkKcg7wEJeGKrZOkWLEhqws0e2I6rkCOpZQ2bv93ZgfkpLQKD8UJAGsix9oeiWnOb9tTZ9g4PBZns_5l04V08RBXdT-EU3HQhC7X078-Ec8310_zRXH_eHs3n90XLUimgsuSm6CqqiolBaWlszYoJoOuMhpNUzEjqCVpDMSAjqypDWiydUMkAScCd9x1ih-bOg--Xsb4Xo4eUujK17Ae6pS9QTW-hZ7QK6BRtdip2r6JaRW-YuoqP4TvLqYmhb5s85aSPUi_Tcv_S8uj8Z8jtY29wh9sNlpn</addsrcrecordid><sourcetype>Publisher</sourcetype><iscdi>true</iscdi><recordtype>book_chapter</recordtype><pqid>EBC6320223_43_214</pqid></control><display><type>book_chapter</type><title>Angular Momenta</title><source>Ebook Central Perpetual and DDA</source><creator>Wan, K. Kong</creator><contributor>Wan, K. Kong</contributor><creatorcontrib>Wan, K. Kong ; Wan, K. Kong</creatorcontrib><description>Q36(1) The spherical harmonics Y
1,1(θ, φ), Y
1,0(θ, φ), Y
1,−1(θ, φ) are given by Eqs. (16.65) to (16.67). In Cartesian coordinates, these functions are given by
1
Y
1
,
−
1
(
x
,
y
,
z
)
=
3
8
π
x
−
i
y
r
,
Y
1
,
0
(
x
,
y
,
z
)
=
3
4
π
z
r
,
Y
1
,
1
(
x
,
y
,
z
)
=
−
3
8
π
x
+
i
y
r
.
(a) Using the expression in Cartesian coordinates for the quantised orbital angular momentum operator
L
^
c
z
in Eq. (27.86), verify by explicit calculations that Y
1,1(x, y, z), Y
1,0(x, y, z)and Y
1,-1(x, y, z) are the eigenfunctions of
L
^
c
z
corresponding to eigenvalues ħ, 0 and –ħ.
(b) Consider the following coordinate transformations:
x
→
z
′
,
y
→
x
′
,
z
→
y
′
.
(1)
202Show that the component of the orbital angular momentum operator along the z′-direction is the same as that along the x-direction, i.e., show that
L
^
c
z
′
=
L
^
c
z
.
(2) Show that the simultaneous eigenfunctions of
L
^
c
2
and
L
^
c
x
corresponding to eigenvalues of
L
^
c
2
equal to 2ħ
2are given in Cartesian coordinates by
X
1
,
−
1
(
x
,
y
,
z
)
=
3
8
π
y
−
i
z
r
,
X
1
,
0
(
x
,
y
,
z
)
=
3
4
π
x
r
,
X
1
,
1
(
x
,
y
,
z
)
=
−
3
8
π
y
+
i
z
r
.
SQ36(1)(a) In Cartesian coordinates we have
2
:
p
^
x
Y
→
1
,
−
1
:
=
−
i
ℏ
3
8
π
∂
∂
x
x
−
i
y
r
=
−
i
ℏ
3
8
π
(
1
r
−
x
−
i
y
r
2
∂
∂
x
r
)
=
−
i
ℏ
3
8
π
(
1
r
−
x
−
i
y
r
2
x
r
)
=
−
i
ℏ
3
8
π
1
r
3
(
r
2
−
x
2
+
i
x
y
)
.
p
^
y
Y
→
1
,
−
1
:
=
−
i
ℏ
3
8
π
∂
∂
y
x
−
i
y
r
=
−
i
ℏ
3
8
π
(
−
i
r
−
x
−
i
y
r
2
∂
∂
y
r
)
=
−
i
ℏ
3
8
π
(
−
i
r
−
x
−
i
y
r
2
y
r
)
=
−
i
ℏ
3
8
π
1
r
3
(
−
i
r
2
−
x
y
+
i
y
2
)
.
L
^
z
Y
→
1
,
−
1
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
−
1
=
−
i
ℏ
3
8
π
1
r
3
(
(
−
i
x
r
2
−
x
2
y
+
i
x
y
2
)
−
(
y
r
2
−
y
x
2
+
i
x
y
2
)
)
=
−
i
ℏ
3
8
π
1
r
3
(
−
i
x
r
2
−
y
r
2
)
=
−
ℏ
3
8
π
1
r
{
x
−
i
y
}
=
−
ℏ
Y
1
,
−
1
.
203Next we have
p
^
x
Y
→
1
,
0
:
=
−
i
ℏ
3
4
π
∂
∂
x
z
r
=
−
i
ℏ
3
4
π
(
−
z
r
2
∂
∂
x
r
)
=
i
ℏ
3
4
π
z
x
r
3
.
p
^
y
Y
→
1
,
0
:
=
−
i
ℏ
3
4
π
∂
∂
y
z
r
=
−
i
ℏ
3
4
π
(
−
z
r
2
∂
∂
y
r
)
=
i
ℏ
3
4
π
z
y
r
3
.
L
^
z
Y
→
1
,
0
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
0
=
x
(
i
ℏ
3
4
π
z
y
r
3
)
−
y
(
i
ℏ
3
4
π
z
x
r
3
)
=
0.
Finally we have
p
^
x
Y
→
1
,
1
:
=
i
ℏ
3
8
π
∂
∂
x
x
+
i
y
r
=
−
i
ℏ
3
8
π
(
1
r
−
x
+
i
y
r
2
∂
∂
x
r
)
=
i
ℏ
3
8
π
(
1
r
−
x
+
i
y
r
2
x
r
)
=
i
ℏ
3
8
π
1
r
3
(
r
2
−
x
2
−
i
x
y
)
,
p
^
y
Y
→
1
,
1
:
=
i
ℏ
3
8
π
∂
∂
y
x
+
i
y
r
=
i
ℏ
3
8
π
(
i
r
−
x
+
i
y
r
2
∂
∂
y
r
)
=
i
ℏ
3
8
π
(
i
r
−
x
+
i
y
r
2
y
r
)
=
i
ℏ
3
8
π
1
r
3
(
i
r
2
−
x
y
−
i
y
2
)
,
L
^
z
Y
→
1
,
1
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
1
=
i
ℏ
3
8
π
1
r
3
{
(
i
x
r
2
−
x
2
y
−
i
x
y
2
)
=
−
(
y
r
2
−
y
x
2
−
i
x
y
2
)
}
=
i
ℏ
3
8
π
1
r
3
{
i
x
r
2
−
y
r
2
}
=
−
ℏ
3
8
π
1
r
(
x
+
i
y
)
=
ℏ
Y
1
,
1
.
204
SQ36(1)(b)
(1) Consider a transformation from the original coordinate system (x, y, z) to a new coordinate system (x′, y′, z′):
x
→
z
′
,
y
→
x
′
,
z
→
y
′
.
We can see that the new z′-axis coincides with the original x-axis, and that
r
=
x
2
+
y
2
+
z
2
=
x
′
2
+
y
′
2
+
z
′
2
=
r
′
.
Physically we expect the angular momentum about the x-axis to be the same as the angular momentum about the z′-axis. We can confirm this mathematically by showing that
L
^
x
=
y
p
^
z
−
z
p
^
y
=
−
i
ℏ
(
y
∂
∂
z
−
z
∂
∂
y
)
is equal to
L
^
z
′
=
x
′
p
^
y
′
−
y
′
p
^
x
′
=
−
i
ℏ
(
x
′
∂
∂
y
′
−
y
′
∂
∂
x
′
)
.
This is obvious since x′ = y, y′ = z. It follows that the eigenfunctions and eigenvalues of
L
^
x
are the same in terms of the dashed coordinates as that of
L
^
z
′
.
(2) We know the eigenfunctions
Y
ℓ
,
m
ℓ
′
(
x
′
,
y
′
,
z
′
)
and eigenvalues of
L
^
z
′
in the transformed coordinates (x′, y′, z′) as they are of the same form as those of the eigenfunctions and eigenvalues of
L
^
z
in the original coordinates, i.e., we have
Y
1
,
−
1
′
(
x
′
,
y
′
,
z
′
)
=
3
8
π
x
′
−
i
y
′
r
′
,
Y
1
,
0
′
(
x
′
,
y
′
,
z
′
)
=
3
4
π
z
′
r
′
Y
1
,
1
′
(
x
′
,
y
′
,
z
′
)
=
−
3
8
π
x
′
+
i
y
′
r
′
.
205Since
L
^
z
′
=
L
^
x
these are also eigenfunctions of
L
^
x
Written in terms of the original coordinates these functions become
Y
1
,
−
1
′
=
3
8
π
y
−
i
z
r
,
Y
1
,
0
′
=
3
4
π
x
r
,
Y
1
,
1
′
=
−
3
8
π
y
+
i
z
r
.
We can conclude that the required eigenfunctions
L
^
x
denoted more conveniently by X
1,1, X
1,0, X
1,−1 are
X
1
,
−
1
=
3
8
π
y
−
i
z
r
,
X
1
,
0
=
3
4
π
x
r
,
X
1
,
1
=
−
3
8
π
y
+
i
z
r
.
Q36(2) Suppose
L
^
c
2
and
L
^
c
z
are measured giving the eigenvalues 2ħ
2 and –ħ, respectively. A measurement of
L
^
c
x
is then made. What are the possible results of the measurement of
L
^
c
x
? Find the probability of each of these possible results.</description><edition>1</edition><identifier>ISBN: 9789814800723</identifier><identifier>ISBN: 9814800724</identifier><identifier>EISBN: 9781000022568</identifier><identifier>EISBN: 1000022102</identifier><identifier>EISBN: 9781000022339</identifier><identifier>EISBN: 1000022331</identifier><identifier>EISBN: 1000022560</identifier><identifier>EISBN: 0429296479</identifier><identifier>EISBN: 9780429296475</identifier><identifier>EISBN: 9781000022100</identifier><identifier>DOI: 10.1201/9780429296475-36</identifier><identifier>OCLC: 1195448957</identifier><identifier>LCCallNum: QA76.3 .K664 2021</identifier><language>eng</language><publisher>United Kingdom: Jenny Stanford Publishing</publisher><ispartof>Quantum Mechanics, 2021, p.201-223</ispartof><rights>Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.</rights><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Uhttps://ebookcentral.proquest.com/covers/6320223-l.jpg</thumbnail><linktopdf>$$Uhttps://ebookcentral.proquest.com/lib/munchentech/reader.action?docID=6320223_43_214&ppg=214$$EPDF$$P50$$Gproquest$$H</linktopdf><link.rule.ids>779,780,784,793,27925,79398</link.rule.ids></links><search><contributor>Wan, K. Kong</contributor><creatorcontrib>Wan, K. Kong</creatorcontrib><title>Angular Momenta</title><title>Quantum Mechanics</title><description>Q36(1) The spherical harmonics Y
1,1(θ, φ), Y
1,0(θ, φ), Y
1,−1(θ, φ) are given by Eqs. (16.65) to (16.67). In Cartesian coordinates, these functions are given by
1
Y
1
,
−
1
(
x
,
y
,
z
)
=
3
8
π
x
−
i
y
r
,
Y
1
,
0
(
x
,
y
,
z
)
=
3
4
π
z
r
,
Y
1
,
1
(
x
,
y
,
z
)
=
−
3
8
π
x
+
i
y
r
.
(a) Using the expression in Cartesian coordinates for the quantised orbital angular momentum operator
L
^
c
z
in Eq. (27.86), verify by explicit calculations that Y
1,1(x, y, z), Y
1,0(x, y, z)and Y
1,-1(x, y, z) are the eigenfunctions of
L
^
c
z
corresponding to eigenvalues ħ, 0 and –ħ.
(b) Consider the following coordinate transformations:
x
→
z
′
,
y
→
x
′
,
z
→
y
′
.
(1)
202Show that the component of the orbital angular momentum operator along the z′-direction is the same as that along the x-direction, i.e., show that
L
^
c
z
′
=
L
^
c
z
.
(2) Show that the simultaneous eigenfunctions of
L
^
c
2
and
L
^
c
x
corresponding to eigenvalues of
L
^
c
2
equal to 2ħ
2are given in Cartesian coordinates by
X
1
,
−
1
(
x
,
y
,
z
)
=
3
8
π
y
−
i
z
r
,
X
1
,
0
(
x
,
y
,
z
)
=
3
4
π
x
r
,
X
1
,
1
(
x
,
y
,
z
)
=
−
3
8
π
y
+
i
z
r
.
SQ36(1)(a) In Cartesian coordinates we have
2
:
p
^
x
Y
→
1
,
−
1
:
=
−
i
ℏ
3
8
π
∂
∂
x
x
−
i
y
r
=
−
i
ℏ
3
8
π
(
1
r
−
x
−
i
y
r
2
∂
∂
x
r
)
=
−
i
ℏ
3
8
π
(
1
r
−
x
−
i
y
r
2
x
r
)
=
−
i
ℏ
3
8
π
1
r
3
(
r
2
−
x
2
+
i
x
y
)
.
p
^
y
Y
→
1
,
−
1
:
=
−
i
ℏ
3
8
π
∂
∂
y
x
−
i
y
r
=
−
i
ℏ
3
8
π
(
−
i
r
−
x
−
i
y
r
2
∂
∂
y
r
)
=
−
i
ℏ
3
8
π
(
−
i
r
−
x
−
i
y
r
2
y
r
)
=
−
i
ℏ
3
8
π
1
r
3
(
−
i
r
2
−
x
y
+
i
y
2
)
.
L
^
z
Y
→
1
,
−
1
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
−
1
=
−
i
ℏ
3
8
π
1
r
3
(
(
−
i
x
r
2
−
x
2
y
+
i
x
y
2
)
−
(
y
r
2
−
y
x
2
+
i
x
y
2
)
)
=
−
i
ℏ
3
8
π
1
r
3
(
−
i
x
r
2
−
y
r
2
)
=
−
ℏ
3
8
π
1
r
{
x
−
i
y
}
=
−
ℏ
Y
1
,
−
1
.
203Next we have
p
^
x
Y
→
1
,
0
:
=
−
i
ℏ
3
4
π
∂
∂
x
z
r
=
−
i
ℏ
3
4
π
(
−
z
r
2
∂
∂
x
r
)
=
i
ℏ
3
4
π
z
x
r
3
.
p
^
y
Y
→
1
,
0
:
=
−
i
ℏ
3
4
π
∂
∂
y
z
r
=
−
i
ℏ
3
4
π
(
−
z
r
2
∂
∂
y
r
)
=
i
ℏ
3
4
π
z
y
r
3
.
L
^
z
Y
→
1
,
0
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
0
=
x
(
i
ℏ
3
4
π
z
y
r
3
)
−
y
(
i
ℏ
3
4
π
z
x
r
3
)
=
0.
Finally we have
p
^
x
Y
→
1
,
1
:
=
i
ℏ
3
8
π
∂
∂
x
x
+
i
y
r
=
−
i
ℏ
3
8
π
(
1
r
−
x
+
i
y
r
2
∂
∂
x
r
)
=
i
ℏ
3
8
π
(
1
r
−
x
+
i
y
r
2
x
r
)
=
i
ℏ
3
8
π
1
r
3
(
r
2
−
x
2
−
i
x
y
)
,
p
^
y
Y
→
1
,
1
:
=
i
ℏ
3
8
π
∂
∂
y
x
+
i
y
r
=
i
ℏ
3
8
π
(
i
r
−
x
+
i
y
r
2
∂
∂
y
r
)
=
i
ℏ
3
8
π
(
i
r
−
x
+
i
y
r
2
y
r
)
=
i
ℏ
3
8
π
1
r
3
(
i
r
2
−
x
y
−
i
y
2
)
,
L
^
z
Y
→
1
,
1
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
1
=
i
ℏ
3
8
π
1
r
3
{
(
i
x
r
2
−
x
2
y
−
i
x
y
2
)
=
−
(
y
r
2
−
y
x
2
−
i
x
y
2
)
}
=
i
ℏ
3
8
π
1
r
3
{
i
x
r
2
−
y
r
2
}
=
−
ℏ
3
8
π
1
r
(
x
+
i
y
)
=
ℏ
Y
1
,
1
.
204
SQ36(1)(b)
(1) Consider a transformation from the original coordinate system (x, y, z) to a new coordinate system (x′, y′, z′):
x
→
z
′
,
y
→
x
′
,
z
→
y
′
.
We can see that the new z′-axis coincides with the original x-axis, and that
r
=
x
2
+
y
2
+
z
2
=
x
′
2
+
y
′
2
+
z
′
2
=
r
′
.
Physically we expect the angular momentum about the x-axis to be the same as the angular momentum about the z′-axis. We can confirm this mathematically by showing that
L
^
x
=
y
p
^
z
−
z
p
^
y
=
−
i
ℏ
(
y
∂
∂
z
−
z
∂
∂
y
)
is equal to
L
^
z
′
=
x
′
p
^
y
′
−
y
′
p
^
x
′
=
−
i
ℏ
(
x
′
∂
∂
y
′
−
y
′
∂
∂
x
′
)
.
This is obvious since x′ = y, y′ = z. It follows that the eigenfunctions and eigenvalues of
L
^
x
are the same in terms of the dashed coordinates as that of
L
^
z
′
.
(2) We know the eigenfunctions
Y
ℓ
,
m
ℓ
′
(
x
′
,
y
′
,
z
′
)
and eigenvalues of
L
^
z
′
in the transformed coordinates (x′, y′, z′) as they are of the same form as those of the eigenfunctions and eigenvalues of
L
^
z
in the original coordinates, i.e., we have
Y
1
,
−
1
′
(
x
′
,
y
′
,
z
′
)
=
3
8
π
x
′
−
i
y
′
r
′
,
Y
1
,
0
′
(
x
′
,
y
′
,
z
′
)
=
3
4
π
z
′
r
′
Y
1
,
1
′
(
x
′
,
y
′
,
z
′
)
=
−
3
8
π
x
′
+
i
y
′
r
′
.
205Since
L
^
z
′
=
L
^
x
these are also eigenfunctions of
L
^
x
Written in terms of the original coordinates these functions become
Y
1
,
−
1
′
=
3
8
π
y
−
i
z
r
,
Y
1
,
0
′
=
3
4
π
x
r
,
Y
1
,
1
′
=
−
3
8
π
y
+
i
z
r
.
We can conclude that the required eigenfunctions
L
^
x
denoted more conveniently by X
1,1, X
1,0, X
1,−1 are
X
1
,
−
1
=
3
8
π
y
−
i
z
r
,
X
1
,
0
=
3
4
π
x
r
,
X
1
,
1
=
−
3
8
π
y
+
i
z
r
.
Q36(2) Suppose
L
^
c
2
and
L
^
c
z
are measured giving the eigenvalues 2ħ
2 and –ħ, respectively. A measurement of
L
^
c
x
is then made. What are the possible results of the measurement of
L
^
c
x
? Find the probability of each of these possible results.</description><isbn>9789814800723</isbn><isbn>9814800724</isbn><isbn>9781000022568</isbn><isbn>1000022102</isbn><isbn>9781000022339</isbn><isbn>1000022331</isbn><isbn>1000022560</isbn><isbn>0429296479</isbn><isbn>9780429296475</isbn><isbn>9781000022100</isbn><fulltext>true</fulltext><rsrctype>book_chapter</rsrctype><creationdate>2021</creationdate><recordtype>book_chapter</recordtype><recordid>eNpVkMtOAzEMRYMQCCgVW5b8wEAcOw8vqwooEogNrKN0HjAwnZRkCuLvmVI2eGPZ8j26vkKcg7wEJeGKrZOkWLEhqws0e2I6rkCOpZQ2bv93ZgfkpLQKD8UJAGsix9oeiWnOb9tTZ9g4PBZns_5l04V08RBXdT-EU3HQhC7X078-Ec8310_zRXH_eHs3n90XLUimgsuSm6CqqiolBaWlszYoJoOuMhpNUzEjqCVpDMSAjqypDWiydUMkAScCd9x1ih-bOg--Xsb4Xo4eUujK17Ae6pS9QTW-hZ7QK6BRtdip2r6JaRW-YuoqP4TvLqYmhb5s85aSPUi_Tcv_S8uj8Z8jtY29wh9sNlpn</recordid><startdate>2021</startdate><enddate>2021</enddate><creator>Wan, K. Kong</creator><general>Jenny Stanford Publishing</general><scope>FFUUA</scope></search><sort><creationdate>2021</creationdate><title>Angular Momenta</title><author>Wan, K. Kong</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-i1094-9cc9fa2dddc04a250877a294638d6536fd99312b453a49138476e61547ef44013</frbrgroupid><rsrctype>book_chapters</rsrctype><prefilter>book_chapters</prefilter><language>eng</language><creationdate>2021</creationdate><toplevel>online_resources</toplevel><creatorcontrib>Wan, K. Kong</creatorcontrib><collection>ProQuest Ebook Central - Book Chapters - Demo use only</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Wan, K. Kong</au><au>Wan, K. Kong</au><format>book</format><genre>bookitem</genre><ristype>CHAP</ristype><atitle>Angular Momenta</atitle><btitle>Quantum Mechanics</btitle><date>2021</date><risdate>2021</risdate><spage>201</spage><epage>223</epage><pages>201-223</pages><isbn>9789814800723</isbn><isbn>9814800724</isbn><eisbn>9781000022568</eisbn><eisbn>1000022102</eisbn><eisbn>9781000022339</eisbn><eisbn>1000022331</eisbn><eisbn>1000022560</eisbn><eisbn>0429296479</eisbn><eisbn>9780429296475</eisbn><eisbn>9781000022100</eisbn><abstract>Q36(1) The spherical harmonics Y
1,1(θ, φ), Y
1,0(θ, φ), Y
1,−1(θ, φ) are given by Eqs. (16.65) to (16.67). In Cartesian coordinates, these functions are given by
1
Y
1
,
−
1
(
x
,
y
,
z
)
=
3
8
π
x
−
i
y
r
,
Y
1
,
0
(
x
,
y
,
z
)
=
3
4
π
z
r
,
Y
1
,
1
(
x
,
y
,
z
)
=
−
3
8
π
x
+
i
y
r
.
(a) Using the expression in Cartesian coordinates for the quantised orbital angular momentum operator
L
^
c
z
in Eq. (27.86), verify by explicit calculations that Y
1,1(x, y, z), Y
1,0(x, y, z)and Y
1,-1(x, y, z) are the eigenfunctions of
L
^
c
z
corresponding to eigenvalues ħ, 0 and –ħ.
(b) Consider the following coordinate transformations:
x
→
z
′
,
y
→
x
′
,
z
→
y
′
.
(1)
202Show that the component of the orbital angular momentum operator along the z′-direction is the same as that along the x-direction, i.e., show that
L
^
c
z
′
=
L
^
c
z
.
(2) Show that the simultaneous eigenfunctions of
L
^
c
2
and
L
^
c
x
corresponding to eigenvalues of
L
^
c
2
equal to 2ħ
2are given in Cartesian coordinates by
X
1
,
−
1
(
x
,
y
,
z
)
=
3
8
π
y
−
i
z
r
,
X
1
,
0
(
x
,
y
,
z
)
=
3
4
π
x
r
,
X
1
,
1
(
x
,
y
,
z
)
=
−
3
8
π
y
+
i
z
r
.
SQ36(1)(a) In Cartesian coordinates we have
2
:
p
^
x
Y
→
1
,
−
1
:
=
−
i
ℏ
3
8
π
∂
∂
x
x
−
i
y
r
=
−
i
ℏ
3
8
π
(
1
r
−
x
−
i
y
r
2
∂
∂
x
r
)
=
−
i
ℏ
3
8
π
(
1
r
−
x
−
i
y
r
2
x
r
)
=
−
i
ℏ
3
8
π
1
r
3
(
r
2
−
x
2
+
i
x
y
)
.
p
^
y
Y
→
1
,
−
1
:
=
−
i
ℏ
3
8
π
∂
∂
y
x
−
i
y
r
=
−
i
ℏ
3
8
π
(
−
i
r
−
x
−
i
y
r
2
∂
∂
y
r
)
=
−
i
ℏ
3
8
π
(
−
i
r
−
x
−
i
y
r
2
y
r
)
=
−
i
ℏ
3
8
π
1
r
3
(
−
i
r
2
−
x
y
+
i
y
2
)
.
L
^
z
Y
→
1
,
−
1
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
−
1
=
−
i
ℏ
3
8
π
1
r
3
(
(
−
i
x
r
2
−
x
2
y
+
i
x
y
2
)
−
(
y
r
2
−
y
x
2
+
i
x
y
2
)
)
=
−
i
ℏ
3
8
π
1
r
3
(
−
i
x
r
2
−
y
r
2
)
=
−
ℏ
3
8
π
1
r
{
x
−
i
y
}
=
−
ℏ
Y
1
,
−
1
.
203Next we have
p
^
x
Y
→
1
,
0
:
=
−
i
ℏ
3
4
π
∂
∂
x
z
r
=
−
i
ℏ
3
4
π
(
−
z
r
2
∂
∂
x
r
)
=
i
ℏ
3
4
π
z
x
r
3
.
p
^
y
Y
→
1
,
0
:
=
−
i
ℏ
3
4
π
∂
∂
y
z
r
=
−
i
ℏ
3
4
π
(
−
z
r
2
∂
∂
y
r
)
=
i
ℏ
3
4
π
z
y
r
3
.
L
^
z
Y
→
1
,
0
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
0
=
x
(
i
ℏ
3
4
π
z
y
r
3
)
−
y
(
i
ℏ
3
4
π
z
x
r
3
)
=
0.
Finally we have
p
^
x
Y
→
1
,
1
:
=
i
ℏ
3
8
π
∂
∂
x
x
+
i
y
r
=
−
i
ℏ
3
8
π
(
1
r
−
x
+
i
y
r
2
∂
∂
x
r
)
=
i
ℏ
3
8
π
(
1
r
−
x
+
i
y
r
2
x
r
)
=
i
ℏ
3
8
π
1
r
3
(
r
2
−
x
2
−
i
x
y
)
,
p
^
y
Y
→
1
,
1
:
=
i
ℏ
3
8
π
∂
∂
y
x
+
i
y
r
=
i
ℏ
3
8
π
(
i
r
−
x
+
i
y
r
2
∂
∂
y
r
)
=
i
ℏ
3
8
π
(
i
r
−
x
+
i
y
r
2
y
r
)
=
i
ℏ
3
8
π
1
r
3
(
i
r
2
−
x
y
−
i
y
2
)
,
L
^
z
Y
→
1
,
1
:
=
(
x
p
^
y
−
y
p
^
x
)
Y
1
,
1
=
i
ℏ
3
8
π
1
r
3
{
(
i
x
r
2
−
x
2
y
−
i
x
y
2
)
=
−
(
y
r
2
−
y
x
2
−
i
x
y
2
)
}
=
i
ℏ
3
8
π
1
r
3
{
i
x
r
2
−
y
r
2
}
=
−
ℏ
3
8
π
1
r
(
x
+
i
y
)
=
ℏ
Y
1
,
1
.
204
SQ36(1)(b)
(1) Consider a transformation from the original coordinate system (x, y, z) to a new coordinate system (x′, y′, z′):
x
→
z
′
,
y
→
x
′
,
z
→
y
′
.
We can see that the new z′-axis coincides with the original x-axis, and that
r
=
x
2
+
y
2
+
z
2
=
x
′
2
+
y
′
2
+
z
′
2
=
r
′
.
Physically we expect the angular momentum about the x-axis to be the same as the angular momentum about the z′-axis. We can confirm this mathematically by showing that
L
^
x
=
y
p
^
z
−
z
p
^
y
=
−
i
ℏ
(
y
∂
∂
z
−
z
∂
∂
y
)
is equal to
L
^
z
′
=
x
′
p
^
y
′
−
y
′
p
^
x
′
=
−
i
ℏ
(
x
′
∂
∂
y
′
−
y
′
∂
∂
x
′
)
.
This is obvious since x′ = y, y′ = z. It follows that the eigenfunctions and eigenvalues of
L
^
x
are the same in terms of the dashed coordinates as that of
L
^
z
′
.
(2) We know the eigenfunctions
Y
ℓ
,
m
ℓ
′
(
x
′
,
y
′
,
z
′
)
and eigenvalues of
L
^
z
′
in the transformed coordinates (x′, y′, z′) as they are of the same form as those of the eigenfunctions and eigenvalues of
L
^
z
in the original coordinates, i.e., we have
Y
1
,
−
1
′
(
x
′
,
y
′
,
z
′
)
=
3
8
π
x
′
−
i
y
′
r
′
,
Y
1
,
0
′
(
x
′
,
y
′
,
z
′
)
=
3
4
π
z
′
r
′
Y
1
,
1
′
(
x
′
,
y
′
,
z
′
)
=
−
3
8
π
x
′
+
i
y
′
r
′
.
205Since
L
^
z
′
=
L
^
x
these are also eigenfunctions of
L
^
x
Written in terms of the original coordinates these functions become
Y
1
,
−
1
′
=
3
8
π
y
−
i
z
r
,
Y
1
,
0
′
=
3
4
π
x
r
,
Y
1
,
1
′
=
−
3
8
π
y
+
i
z
r
.
We can conclude that the required eigenfunctions
L
^
x
denoted more conveniently by X
1,1, X
1,0, X
1,−1 are
X
1
,
−
1
=
3
8
π
y
−
i
z
r
,
X
1
,
0
=
3
4
π
x
r
,
X
1
,
1
=
−
3
8
π
y
+
i
z
r
.
Q36(2) Suppose
L
^
c
2
and
L
^
c
z
are measured giving the eigenvalues 2ħ
2 and –ħ, respectively. A measurement of
L
^
c
x
is then made. What are the possible results of the measurement of
L
^
c
x
? Find the probability of each of these possible results.</abstract><cop>United Kingdom</cop><pub>Jenny Stanford Publishing</pub><doi>10.1201/9780429296475-36</doi><oclcid>1195448957</oclcid><tpages>23</tpages><edition>1</edition></addata></record> |
| fulltext | fulltext |
| identifier | ISBN: 9789814800723 |
| ispartof | Quantum Mechanics, 2021, p.201-223 |
| issn | |
| language | eng |
| recordid | cdi_proquest_ebookcentralchapters_6320223_43_214 |
| source | Ebook Central Perpetual and DDA |
| title | Angular Momenta |
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