ϕ 2p xþiy
À
Á Y
1
1 θ, ϕ
ð
Þ e
R
2
ð Þ
1 r
ð Þ ¼ À
1
8
ffiffiffi
π
p a
À
3
2
r
a
e
À
r
2a sin θe
iϕ
¼ À
1
8
ffiffiffi
π
p a
À
3
2
r
a
e
À
r
2a
x þ iy
r
¼ À
1
8
ffiffiffi
π
p a
À
5
2 e
À
r
2a x þ iy
ð
Þ:
ð3:304Þ
In (3.304), the minus sign comes from the Condon–Shortley phase. Furthermore,
we have
ϕ 2p xÀiy
À
Á Y
À1
1 θ, ϕ
ð
Þ e
R
2
ð Þ
1 r
ð Þ ¼
1
8
ffiffiffi
π
p a
À
3
2
r
a
e
À
r
2a sin θe
Àiϕ
¼
1
8
ffiffiffi
π
p a
À
3
2
r
a
e
À
r
2a
x À iy
r
¼
1
8
ffiffiffi
π
p a
À
5
2 e
À
r
2a x À iy
ð
Þ:
ð3:305Þ
Notice that the above notations ϕ(2p x + iy ) and ϕ(2p x À iy ) differ from the custom
that uses, e.g., ϕ(2p x ) and ϕ(2p y ). We will come back to this point in Sect. 4.3.
References
1. Schiff LI (1955) Quantum mechanics, 2nd edn. McGraw-Hill, New York
2. Arfken GB (1970) Mathematical methods for physicists, 2nd edn. Academic Press, Waltham
3. Sunakawa S (1991) Quantum mechanics. Iwanami, Tokyo. (in Japanese)
4. Byron FW Jr, Fuller RW (1992) Mathematics of classical and quantum physics. Dover,
New York
5. Dennery P, Krzywicki A (1996) Mathematics for physicists. Dover, New York
6. Riley KF, Hobson MP, Bence SJ (2006) Mathematical methods for physics and engineering, 3rd
edn. Cambridge University Press, Cambridge
7. Arfken GB, Weber HJ, Harris FE (2013) Mathematical methods for physicists, 7th edn. Academic Press, Waltham
8. Lebedev NN (1972) Special functions and their applications. Dover, New York
9. Stakgold I (1998) Green’s functions and boundary value problems, 2nd edn. Wiley, New York
References
123
À
Á Y
1
1 θ, ϕ
ð
Þ e
R
2
ð Þ
1 r
ð Þ ¼ À
1
8
ffiffiffi
π
p a
À
3
2
r
a
e
À
r
2a sin θe
iϕ
¼ À
1
8
ffiffiffi
π
p a
À
3
2
r
a
e
À
r
2a
x þ iy
r
¼ À
1
8
ffiffiffi
π
p a
À
5
2 e
À
r
2a x þ iy
ð
Þ:
ð3:304Þ
In (3.304), the minus sign comes from the Condon–Shortley phase. Furthermore,
we have
ϕ 2p xÀiy
À
Á Y
À1
1 θ, ϕ
ð
Þ e
R
2
ð Þ
1 r
ð Þ ¼
1
8
ffiffiffi
π
p a
À
3
2
r
a
e
À
r
2a sin θe
Àiϕ
¼
1
8
ffiffiffi
π
p a
À
3
2
r
a
e
À
r
2a
x À iy
r
¼
1
8
ffiffiffi
π
p a
À
5
2 e
À
r
2a x À iy
ð
Þ:
ð3:305Þ
Notice that the above notations ϕ(2p x + iy ) and ϕ(2p x À iy ) differ from the custom
that uses, e.g., ϕ(2p x ) and ϕ(2p y ). We will come back to this point in Sect. 4.3.
References
1. Schiff LI (1955) Quantum mechanics, 2nd edn. McGraw-Hill, New York
2. Arfken GB (1970) Mathematical methods for physicists, 2nd edn. Academic Press, Waltham
3. Sunakawa S (1991) Quantum mechanics. Iwanami, Tokyo. (in Japanese)
4. Byron FW Jr, Fuller RW (1992) Mathematics of classical and quantum physics. Dover,
New York
5. Dennery P, Krzywicki A (1996) Mathematics for physicists. Dover, New York
6. Riley KF, Hobson MP, Bence SJ (2006) Mathematical methods for physics and engineering, 3rd
edn. Cambridge University Press, Cambridge
7. Arfken GB, Weber HJ, Harris FE (2013) Mathematical methods for physicists, 7th edn. Academic Press, Waltham
8. Lebedev NN (1972) Special functions and their applications. Dover, New York
9. Stakgold I (1998) Green’s functions and boundary value problems, 2nd edn. Wiley, New York
References
123