8.1 The hydrogen atom solutions

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1 8.1 The hydrogen atom solutions Slides: Video Separating for the radial equation Text reference: Quantum Mechanics for Scientists and Engineers Section 10.4 (up to Solution of the hydrogen radial wavefunction ).

2 The hydrogen atom solutions Separating for the radial equation Quantum mechanics for scientists and engineers David Miller

3 Internal states of the hydrogen atom We start with the equation for the relative motion of electron and proton r Vr U EHU r r We use the spherical symmetry of this equation and change to spherical polar coordinates From now on, we drop the subscript r in the operator

4 Internal states of the hydrogen atom In spherical polar coordinates, we have r sin r r r r sin sin where the term in square brackets is the operator ˆ, L / we introduced in discussing angular momentum Knowing the solutions to the angular momentum problem we propose the separation U r RrY,

5 Internal states of the hydrogen atom The mathematics is simpler using the form 1 U r ry, r where, obviously r rr r This choice gives a convenient simplification of the radial derivatives 1 r 1 r r r r r r r r

6 Internal states of the hydrogen atom Hence the Schrödinger equation becomes 1 r r 1 ˆ r Y, LY 3, Y, Vr r r r r Dividing by ry, /r and rearranging, we have r 3 E H 1 r ry, r 1 1 r E ˆ H V r LY, r r Y,

7 Internal states of the hydrogen atom In r r 1 1 r E ˆ H V r LY, ll1 r r Y, in the usual manner for a separation argument the left hand side depends only on r and the right hand side depends only on and so both sides must be equal to a constant We already know what that constant is explicitly i.e., we already know that LY ˆ lm, ll1 Ylm, so that the constant is l l1

8 Internal states of the hydrogen atom Hence, in addition to the ˆL eigenequation which we had already solved from our separation above, we also have r r r E H V r l l1 r r or, rearranging d r ll1 Vr r EH r dr r which we can write as an ordinary differential equation All the functions and derivatives are in one variable, r

9 Internal states of the hydrogen atom Hence we have mathematical equation d r ll1 Vr r EH r dr r for this radial part of the wavefunction which looks like a Schrödinger wave equation with an additional effective potential energy term of the form ll1 r

10 Central potentials Note incidentally that though here we have a specific form for in our assumed Coulomb potential e V re rp 4 r r o e p Vr the above separation works for any potential that is only a function of r sometimes known as a central potential

11 Central potentials The precise form of the equation d r l l1 Vr r EH r dr r will be different for different central potentials but the separation remains We can still separate out the ˆL angular momentum eigenequation with the spherical harmonic solutions

12 Central potentials Since a reasonable first approximation for more complicated atoms is to say that the potential is still approximately central approximately independent of angle we can continue to use the spherical harmonics as the first approximation to the angular form of the orbitals and use the hydrogen atom labels for them e.g., s, p, d, f, etc.

14 8.1 The hydrogen atom solutions Slides: Video Radial equation solutions Text reference: Quantum Mechanics for Scientists and Engineers Sections 10.4 starting with Solution of the hydrogen radial wavefunction, and 10.5 Note: The section Text reference: Quantum Mechanics for Scientists and Engineers Section 10.4 contains the complete mathematical details for solving the radial equation in the hydrogen atom problem. For this course, not all those details are required and they are consequently not all covered in the online lectures, so the additional detail, in particular on power series solutions in section Text reference: Quantum Mechanics for Scientists and Engineers Section 10.4, is optional for the student.

15 The hydrogen atom solutions Radial equation solutions Quantum mechanics for scientists and engineers David Miller

16 Radial equation solutions Using a separation of the hydrogen atom wavefunction solutions into radial and angular parts Ur RrY, and rewriting the radial part using r rrr we obtained the radial equation d r e l l1 r EH r dr 4or r where we know l is 0 or any positive integer

17 Radial equation solutions We now choose to write our energies in the form Ry EH n where n for now is just an arbitrary real number We define a new distance unit s r where the parameter is E na o H

18 Radial equation solutions We therefore obtain an equation d ll1 n 1 0 ds s s 4 Then we write l1 s s L s exp s/ so we get d L dl s s l 1 n l 1L 0 ds ds

19 Radial equation solutions The technique to solve this equation d L dl s s l 1 n l 1 L 0 ds ds is to propose a power series in s The power series will go on forever and hence the function will grow arbitrarily unless it terminates at some finite power which requires that n is an integer, and nl1

20 Radial equation solutions The normalizable solutions of d L dl s s l 1 n l 1 L 0 ds ds then become the finite power series known as the associated Laguerre polynomials n l 1 l1 q n l! q Ln l 1 s 1 s nlq1! ql1! q0 or equivalently p 1 q0 q p j! j Lp s s p q! j q! q! q

21 Radial equation solutions Now we can work back to construct the whole solution l1 In our definition s s Lsexp s/ we now insert the associated Laguerre polynomials l1 l1 ss Ln l 1 sexp s/ where s ( / nao ) r Since our radial solution was r rrr we now have 1 l1 l1 Rr naos/ s Ln l 1 sexp s/ r l l1 sl 1 sexp s/ n l

22 Radial equation solutions - normalization We formally introduce a normalization coefficient A so 1 l l1 Rr naos/ s Ln l 1 sexp s/ A The full normalization integral of the wavefunction Ur RrY, would be 1 R ry, rsindddr r000 but we have already normalized the spherical harmonics so we are left with the radial normalization

23 Radial equation solutions - normalization Radial normalization would be We could show 0 1 R r r dr so the normalized radial wavefunction becomes 0 l l1 n n l! s Ln l 1 s expss ds n l1! R r 1/ 3 l nl1! r l1 r r Ln l 1 exp nn l! nao nao nao na o

24 Hydrogen atom radial wavefunctions We write the wavefunctions using the Bohr radius a o as the unit of radial distance so we have a dimensionless radial distance r/ ao and we introduce the subscripts n - the principal quantum number, and l - the angular momentum quantum number to index the various functions R n,l

25 Radial wavefunctions - n = 1 Principal quantum number n = 1 Angular momentum quantum number l = 0 R1,0 exp R R1, Radius

26 Radial wavefunctions - n = l = 0 R,0 4 exp / l = 1 6 R,1 1 exp / R R,0 R, Radius

27 Radial wavefunctions - n = 3 R l = 0 3, exp /3 l = 1 6 R3,1 4 exp / l = 30 R3, 115 exp / 3 R R3,0 R3,1 R3, Radius

28 Hydrogen orbital probability density z x - z cross-section at y = 0 x 1s n = 1 l = 0 m = MMZ

29 Hydrogen orbital probability density z x - z cross-section at y = 0 x logarithmic 1. MMZl intensity scale 1s n = 1 l = 0 m = 0

30 Hydrogen orbital probability density z x - z cross-section at y = 0 x s n = l = 0 m = MMZ

31 Hydrogen orbital probability density z x - z cross-section at y = 0 x logarithmic 1. MMZl intensity scale s n = l = 0 m = 0

32 Hydrogen orbital probability density z x - z cross-section at y = 0 x p n = l = 1 m = 0 MMZ

33 Hydrogen orbital probability density z x - z cross-section at y = 0 x 3s n = 3 l = 0 m = 0.3 MMZ

34 Hydrogen orbital probability density z x - z cross-section at y = 0 x logarithmic 1. MMZl intensity scale 3s n = 3 l = 0 m = 0

35 Hydrogen orbital probability density z x - z cross-section at y = 0 x 3p n = 3 l = 1 m = 0 MMZ

36 Hydrogen orbital probability density z x - z cross-section at y = 0 x logarithmic 1.5 MMZl intensity scale 3p n = 3 l = 1 m = 0

37 Hydrogen orbital probability density z x - z cross-section at y = 0 x 3d n = 3 l = m = 0 4 MMZ

38 Hydrogen orbital probability density z x - z cross-section at y = 0 x 3d n = 3 l = m = 1 10 MMZ

39 Behavior of the complete hydrogen solutions (i) The overall size of the wavefunctions becomes larger with larger n (ii) The number of zeros in the wavefunction is n 1 The radial wavefunctions have n l 1 zeros and the spherical harmonics have l nodal circles The radial wavefunctions appear to have an additional zero at r = 0 for all l 1, but this is already counted because the spherical harmonics have at least one nodal circle for all l 1 which already gives a zero as r 0 in these cases

40 Behavior of the complete hydrogen solutions In summary of the quantum numbers for the so-called principal quantum number n 1,,3, and l n1 We already deduced that l is a positive or zero integer We also now know the eigenenergies Given the possible values for n Ry EH n Note the energy does not depend on l (or m)

1 r 2 sin 2 θ. This must be the case as we can see by the following argument + L2

1 r 2 sin 2 θ. This must be the case as we can see by the following argument + L2 PHYS 4 3. The momentum operator in three dimensions is p = i Therefore the momentum-squared operator is [ p 2 = 2 2 = 2 r 2 ) + r 2 r r r 2 sin θ We notice that this can be written as sin θ ) + θ θ r 2