Modue "Atomic physics and atomic stuctue" Lectue 7 Quantum Mechanica teatment of One-eecton atoms Page 1 Objectives In this ectue, we wi appy the Schodinge Equation to the simpe system Hydogen and compae the outcome of the Quantum Theoy with the expeimenta esuts obtained fom diffeent expeiments. We wi aso get to know about the wavefunction and its use in deveoping the concept of the stuctue of atoms.
Page Why hydogen? One eecton atom and the simpest bound system in natue Refeing to the adjacent figue -1, the poton (+ve Chage) is at the cente and the eecton (-ve chage) is moving aound the cente. The attactive foce between them is Cooumbic in natue and the potentia is Z V = V ( x, y, z) = 4πε 0 Ze x + y + z x z y Y.Equation -1 Whee Z is the chage of the nuceus, Z = 1 fo hydogen. X Figue - 1 Since it is a two body pobem, we convet it to one body pobem by intoducing educed mass mm µ = m+ M Whee m and M ae the masses of the eecton and poton, espectivey. So, the tota enegy of the system = kinetic enegy + potentia engy 1 x + p y + p z ) + V ( x, y, z) = ( p µ E. Equation - Hee, p x, p y, and p z ae the inea momenta and E is the tota enegy.
Page 3 Fo the quantum mechanica teatment, we wi convet the cassica dynamica quantities (p x, p y, and p z ) to its coesponding quantum mechanica opeatos px i, py i, pz i x y z E i t Substituting in Equation, we get + + V + ( xyz,, ) = i. Equation -3 µ x y z t Hee, we intoduce the WAVEFUNTION to epesent the eecton. ( xyzt,,, ) Ψ=Ψ. Equation -4 The wavefunction contains the infomation about the position and the time evoution of the eectonic motion. We wi undestand moe about this ate. So opeating equation -, on the wavefuntion, ( xyzt,,, ) Ψ=Ψ, ( ) ( ) ( ) ( ) Ψ xyzt,,, Ψ xyzt,,, Ψ xyzt,,, Ψ xyzt,,, + + + V( xyz,, ) Ψ ( xyzt,,, ) = i µ x y z t Ψ Ψ+ VΨ= i µ t. Equation -5 is the Lapacian opeato. x y z = + + This Equation -5 is known as Time dependent Schodinge Equation.
Page 4 Since the potentia V ( x, y, z) does not depend on time t, and we ae inteested to evauate the enegy of the stationay (time independent) states, we can take this wavefunction as the expicit dependence of time such as iet/ ( xyzt,,, ) ψ ( xyze,, ) Ψ = Substituting this to Equation -5, we get µ + = ( xyz,, ) V ( xyz,, ) E ( xyz,, ) ψ ψ ψ This is the Time Independent Schodinge Equation.. Equation -6 Now we have to evauate the stationay state enegies ( E ) of the eecton in this system by soving this equation to expain the obseved specta of hydogen. Since the potentia is spheicay symmetic Ze Ze V ( x, y, z) = = = V ( ) 4πε x + y + z 4πε 0 0 Z We can convet Equation -6 to its spheica poa coodinate (Figue -) fom, µ + = (,, ) V ( ) (,, ) E (,, ) ψ θφ ψ θφ ψ θφ. Equation -7 Radia distance, Poa ange θ and X x θ z φ y Y Azimutha ange φ Figue - The fom of the Lapacian opeato in Spheica poa coodinates is 1 1 1 = + + sinθ θ θ sin θ φ sinθ
Page 5 Using the sepaation of the vaiabes, we can wite the wavefunction as the poduct fom ( ), Θ( ), Φ ( ) of the independent vaiabes R θ φ ψ(, θφ, ) = R( ) Θ( θ) Φ( φ) Substituting in equation 7, 1 R ΘΦ 1 1 sin R ΘΦ R ΘΦ θ V ( ) R ER µ + + + ΘΦ = ΘΦ sinθ θ θ sin θ φ Caying out the patia diffeentiation, ΘΦ R R Φ Θ sin R Θ Φ θ V ( ) R ER µ + + + ΘΦ = ΘΦ sinθ θ θ sin θ φ Mutipying the this equation by µ RΘΦ and eaanging we get, 1 R 1 Θ 1 Φ µ + sinθ + E V + ( ) = 0 R Θ sinθ θ θ Φ sin θ φ Sepaating the adia ( R( ) ) and angua ( ( θ), ( φ) Θ Φ ) pat 1 d dr µ 1 d dθ 1 d Φ + E V ( ) sinθ R d d = Θ sinθ dθ dθ Φ sin θ dφ Note hee that the patia deivative foms ae conveted to tota deivative fom. Now each equation shoud be equa to a constant, et s take as λ.
Page-6 So, the adia pat is 1 d dr µ + E V ( ) R d d = λ 1 d dr µ λ + ( E V ( ) ) R= 0 d d. Equation -8 And the angua pat is 1 d dθ 1 d Φ sinθ + λ = Θ sinθ dθ dθ Φ sin θ dφ sinθ 1 + = Θ dθ dθ Φ dφ dθ d Φ sinθ λ sin θ. Equation -9 Now, sepaating the poa and the azimutha pat sinθ d 1 sinθ dθ sin d Φ + λ θ = m =. Equation -10 Θ dθ dθ Φ dφ Hee, we have taken that both sides shoud be equa to a constant m. Note that we have thee independent equations fom equation 9 and 10.
Page 7 Now et us fist conside the azimutha pat (Equation -10) 1 d Φ = m Φ dφ 1 d Φ + m Φ= 0 Φ dφ The genea soution of this equation may be witten in the fom 1 im Φ m ( φ ) = e φ... Equation -11 π Which satisfies the othonomaity condition π 0 Φ ( φ) Φ ( φ) dφ = δ * m m' mm ' As the eigen functions must be singe vaued, i.e., Φ(0) = Φ(π) which means e im 0 im π = e And using Eue s fomua 1 = Cos( m π) + isin( m π) This is satisfied ony if m = 0, ±1, ±,.. Theefoe, acceptabe soutions ony exist when m can ony have cetain intege vaues, i.e. it is a quantum numbe. Thus, m is caed the magnetic quantum numbe in spectoscopy because it pays oe when atom inteacts with magnetic fieds.
Page 8 Now we wi discuss the poa pat in Equation 10 Reaanging, we get sinθ d dθ sinθ + λ sin θ = m Θ dθ dθ 1 1 sin d d ( θ ) d Θ m sin θ + λ Θ ( θ) = 0 d sin θ θ θ θ... Equation - Let us intoduce new vaiabe ω = Cosθ, the Equation-1 becomes, ( ω) d d Θ m (1 ) ω + λ Θ ( ω) = 0 dω dω 1 ω... Equation -13 The Equation-13 has singuaities at ω = ± 1, which may be eiminated by having a soution Θ in the fom of Θ= Then Equation-13 becomes / (1 ω ) s u s ω m dω dω 1 ω d u du (1 ω ) ω( s+ 1) + λ+ u = 0 Which can be witten as (1 ω d u du s m ) ( s 1) s s u 0 d ω ω dω λ + + + =... Equation -14 1 ω
Page-9 The ast singua tem in Equation-14 can be emoved by taking s = ± m 0, and then we have (1 ω d u ) ω du ( m 1) [ λ m( m 1) ] u 0 dω + dω + + =... Equation -15 Which is a egua equation and hence its seies soution may be witten as u = aω... Equation -16 = 0 Substituting in Equation-15, yieds the ecusion eation a + ( + m)( + m + 1) λ = a ( + 1)( + ) Requiing that the seies be (Equation-16) be imited by a cetain powe = q, i.e. equiing that it be a poynomia of ode q, we have to intoduce the condition a+ = 0, a 0 which equies λ = ( q+ m)( q+ m + 1) Whee q = 0, 1,, 3,. Now we intoduce = q+ m = 0,1,,3,... (obita quantum numbe) such that m λ = ( + 1)
Page-10 Then Equation-1 becomes 1 sin 17 ( θ ) d sin d Θ ( 1) m θ + + Θ ( θ) = 0 d d sin θ θ θ θ... Equation - This is we known associated Legende diffeentia equation with its soution as associated Legende poynomias m m Θ, = C P ( ω)... Equation -18 m Whee m m C is the nomaization facto and P ( ω) is the associated Legende poynomia defined by + m m / ( 1) m d ω P = (1 ω ) + m dω! m m / d = (1 ω ) P ( ω) m dω Whee P ( ω) is the odinay Legende Poynomia. This expession hods fo negative vaues of m aso P ( + m )! ( ω) = ( 1) P ( ω) ( m )! m m m Fom this we can estabish the ange of vaiation of the azimutha (magnetic) quantum numbe m = 0, ±1, ±, ±3,, ± The associated Legende Poynomias satisfy the othonomiization popety 1 1 ( + m )! P ( ω) P ( ω) dω = δ m m, + 1 ( m )! So, C m = + 1( m )! ( + m )!
Page-11 Then Equation-18 becomes Θ = m, + 1( m )! P ( + m )! m ( ω)... Equation -19 So, we have now two quantum numbes, namey, Obita () and magnetic (m ) = 0, 1,, 3,... m =, + 1,..., 0,..., 1, Depending of the vaue of and m, some of the Associated Legende poynomias ae P P 0 0 = 1 ( θ) ( ) = cosθ P = 1 cos 0 ± 1 1 1 P P = 1 3cos θ P = 1 cos θ cosθ 0 ± 1 = 1 cos θ ± 1/ 1/
Page-1 These functions ae 1.5 1.0 0.5 0.5 45 90 135 180 in Deg
Now, we can have the tota angua pat, fom Equation-11 and -19 Y ( θφ, ) =Θ ( θ) Φ ( φ) m m, m = + 1( m )! P 4 π ( + m )!.Equation -0 m (cos θ ) e im φ
Recap In this ectue, we have eant the quantum mechanica teatment of hydogen atom pobem. To do this, we have used the sepaation of vaiabes method and in this method we have witten the tota wavefunction as a poduct function of the adia pat and the angua pat. Since the couomb inteaction between the nuceus and the eecton is adiay symmetic, we have soved the angua pat sepaatey. Whie soving the angua pat of the wavefunction, we have intoduced two quantum numbes namey, azimutha o obita quantum numbe and magnetic quantum numbe m. (****Check the symbo in the ectue*********) In the next ectue we wi sove the adia pat of the wavefunction and wi constuct the tota wavefunction of the hydogen atom. We wi aso cacuate the enegy eves of the hydogen atom and wi compae it with the obseved spectum.