PHYS 3313 – Section 001 Lecture #17

Wednesday, Nov. 7, 2012 PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

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PHYS 3313 – Section 001Lecture #17

Wednesday, Nov. 7, 2012Dr. Jaehoon Yu

• Solutions for Schrodinger Equation for Hydrogen Atom

• Quantum Numbers• Principal Quantum Number• Orbital Angular Momentum Quantum

Number • Magnetic Quantum Number


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Announcements• Research team member list have been updated on the web!

– Remember the deadline for your research paper is Monday, Nov. 26!!

• Reminder: homework #6– CH6 end of chapter problems: 34, 39, 46, 62 and 65– Due on Monday, Nov. 12, in class

• Reading assignments– CH7.6 and the entire CH8

• Colloquium today– At 4pm, Wednesday, Nov. 7, in SH101– Dr. Nick White of Goddard Space Center, NASA

Wednesday, Nov. 7, 2012 3PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

Group – Research Topic Association


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Research Group Research Topic Presentation Date

1 6 12/5-42 5 12/5-53 7 12/5-14 2 12/3-25 1 12/3-36 9 12/3-57 10 12/3-18 4 12/5-39 3 12/5-210 8 12/3-4

Solution of the Schrödinger Equation for Hydrogen• Substitute ψ into the polar Schrodinger equation and separate the

resulting equation into three equations: R(r), f(θ), and g(φ).Separation of Variables• The derivatives in Schrodinger eq. can be written as

• Substituting them into the polar coord. Schrodinger Eq.

• Multiply both sides by r2 sin2 θ / Rfg

Monday, Nov. 5, 2012 5PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

∂ψ∂r= fg

∂R

∂r

fg

r2∂∂r

r2 ∂R∂r

⎛⎝⎜

⎞⎠⎟+

Rgr2 sinθ

∂∂θ

sinθ∂f∂θ

⎛⎝⎜

⎞⎠⎟+

Rfr2 sin2θ

∂2g∂φ2 +

2μh2 E −V( )Rgf =0

sin2θR

∂∂r

r2 ∂R∂r

⎛⎝⎜

⎞⎠⎟+

sinθf

∂∂θ

sinθ∂f∂θ

⎛⎝⎜

⎞⎠⎟+

1g

∂2g∂φ2 +

2μh2 r2 sin2θ E −V( ) =0

−sin2θ

R

∂

∂rr2∂R

∂r⎛⎝⎜

⎞⎠⎟−2μ

h2r2 sin2θ E −V( ) −

sinθ

f

∂

∂θsinθ

∂f

∂θ⎛⎝⎜

⎞⎠⎟=1

g

∂2g

∂φ2Reorganize

∂ψ∂θ

=Rg∂f

∂θ∂2ψ∂φ2

= Rf∂2g

∂φ2

Solution of the Schrödinger Equation• Only r and θ appear on the left-hand side and only φ

appears on the right-hand side of the equation• The left-hand side of the equation cannot change as φ

changes.• The right-hand side cannot change with either r or θ.• Each side needs to be equal to a constant for the equation to

be true in all cases. Set the constant −mℓ2 equal to the right-

hand side of the reorganized equation

• It is convenient to choose a solution to be .

-------- azimuthal equation


d 2g

dφ2 =−ml2g

eimlφ

Solution of the Schrödinger Equation• satisfies the previous equation for any value of mℓ.• The solution be single valued in order to have a valid solution for

any φ, which requires

• mℓ must be zero or an integer (positive or negative) for this to work• If the sign were positive, the solution would not be normalized.• Now, set the remaining equation equal to −mℓ

2 and divide either side with sin2θ and rearrange it.

• Everything depends on r on the left side and θ on the right side of the equation.Wednesday, Nov. 7, 2012 7PHYS 3313-001, Fall 2012

Dr. Jaehoon Yu

eimlφ

g φ( ) =g φ+ 2π( )

g φ =0( ) =g φ =2π( ) e0 =e2π iml

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R

∂∂r

r2 ∂R∂r

⎛⎝⎜

⎞⎠⎟+

2μr2

h2 E −V( ) =ml2

sin2θ−

1f sinθ

∂∂θ

sinθ∂f∂θ

⎛⎝⎜

⎞⎠⎟

Solution of the Schrödinger Equation• Set each side of the equation equal to constant ℓ(ℓ + 1).

– Radial Equation

– Angular Equation

• Schrödinger equation has been separated into three ordinary second-order differential equations, each containing only one variable.


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R

∂∂r

r2 ∂R∂r

⎛⎝⎜

⎞⎠⎟+

2μr2

h2 E −V( ) =l l +1( ) ⇒ml2

sin2θ

−1

fsinθ∂∂θ

sinθ∂f∂θ

⎛⎝⎜

⎞⎠⎟=ll+1 ()⇒

1

r2d

drr2dR

dr⎛⎝⎜

⎞⎠⎟+

2μh2 E −V −

h2

2μl l +1( )

⎡

⎣⎢

⎤

⎦⎥R=0

1

sinθddθ

sinθdfdθ

⎛⎝⎜

⎞⎠⎟+ l l +1( )−

ml2

sin2θ⎡

⎣⎢

⎤

⎦⎥ f =0

Solution of the Radial Equation• The radial equation is called the associated Laguerre

equation, and the solutions R that satisfy the appropriate boundary conditions are called associated Laguerre functions.

• Assume the ground state has ℓ = 0, and this requires mℓ = 0.

We obtain

• The derivative of yields two terms, we obtain


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r2d

drr2dR

dr⎛⎝⎜

⎞⎠⎟+

2μh2 E −V[ ]R=0

d 2R

dr2+

2r

dRdr

+2μh2 E +

e2

4πε0r⎛

⎝⎜⎞

⎠⎟R=0

r2dR

dr

Solution of the Radial Equation• Let’s try a solution where A is a normalization constant,a0

is a constant with the dimension of length.• Take derivatives of R, we obtain.

• To satisfy this equation for any r, each of the two expressions in parentheses must be zero.

• Set the second parentheses equal to zero and solve for a0.

• Set the first parentheses equal to zero and solve for E.

• Both equal to the Bohr’s resultsWednesday, Nov. 7, 2012 10PHYS 3313-001, Fall 2012

Dr. Jaehoon Yu

R =Ae−r a0

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a02+

2μh2 E

⎛

⎝⎜⎞

⎠⎟+

2μe2

4πε0h2 −

2a0

⎛

⎝⎜⎞

⎠⎟1r

=0

a0 =

4πε0h2

μe2

E =

Bohr’s radius

Ground state energy of the hydrogen atom

−E0 = −13.6eV −

h2

2μa02=

PHYS 3313 – Section 001 Lecture #17

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