7.3 The hydrogen atom

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1 7.3 The hydrogen atom Slides: Video Multiple particle wavefunctions Text reference: Quantum Mechanics for Scientists and Engineers Chapter 10 introduction and Section 10.1

2 The hydrogen atom Multiple particle wavefunctions Quantum mechanics for scientists and engineers David Miller

3 Multiple particle systems How should we tackle this problem of two particles, electron and proton? We start by generalizing the Schrödinger equation writing generally for time-independent problems Ĥ E where now we mean that the Hamiltonian Ĥ is the operator representing the energy of the entire system and is the wavefunction representing the state of the entire system

4 Multiple particle wavefunctions For the hydrogen atom there are two particles the electron and the proton Each of these has a set of coordinates associated with it x e, y e, and z e for the electron and x p, y p, and z p for the proton The wavefunction will therefore in general be a function of all six of these coordinates

6 7.3 The hydrogen atom Slides: Video Solving the hydrogen atom problem Text reference: Quantum Mechanics for Scientists and Engineers Sections (up to Bohr radius and Rydberg energy )

7 The hydrogen atom Solving the hydrogen atom problem Quantum mechanics for scientists and engineers David Miller

8 Hamiltonian for the hydrogen atom The electron and proton each have a mass m e and m p respectively We expect kinetic energy operators associated with each of these masses potential energy from the electrostatic attraction of electron and proton

9 Hamiltonian for the hydrogen atom Hence, the Hamiltonian becomes Hˆ V m m r r where we mean e and similarly for p and re xei yejzek is the position vector of the electron coordinates and similarly for e p e p p e x y z e e e r p

10 Hamiltonian for the hydrogen atom The Coulomb potential energy e V re rp depends on the distance 4 o re rp re rh between the electron and proton coordinates which is important in simplifying the solution The Schrödinger equation can now be written explicitly m e p V e p xe ye ze xp yp zp e mp r r,,,,, E x, y, z, x, y, z e e e p p p

11 Center of mass coordinates The potential here is only a function of re rp the separation of the electron and proton We could choose a new set of six coordinates in which three are the relative positions x xe xp y ye yp z ze zp i.e., a relative position vector r xi yjzk from which we obtain r x y z re What should we choose for the other three coordinates? r p

12 Center of mass coordinates The position R of the center of mass of two masses is the same as the balance point of a light-weight beam with the two masses at opposite ends and so is the weighted average of the positions of the two individual masses mere mprp R M where M is the total mass M m m e p

13 Center of mass coordinates Now we construct the differential operators we need in terms of these coordinates With R XiYjZk then for the new coordinates in the x direction we have mx e e mx p p X x xe xp M and similarly for the y and z directions

14 Center of mass coordinates Using the standard method of changing partial derivatives to new coordinates and fully notating the variables held constant the first derivatives in the x direction become X x me x x X x x M X x e x e x x e x X x X p p p and similarly X x mp x x X x x M X x p x p x x p x X x X e e e

15 Center of mass coordinates The second derivatives become x x x e x e x e x p p p me m e M X x M x x X X X x X x x X and similarly mp mp p x x X X x x X x M X x M x X X x e m e M x X x x e x p x e x p X

16 Center of mass coordinates So dropping the explicit statement of variables held constant 1 1 me mh 1 1 me xe mp xp M X me m p x 1 1 M X x mm e p where is the so-called reduced mass m m e p

17 Center of mass coordinates The same kinds of relations can be written for each of the other Cartesian directions so if we define R and r X Y Z x y z we can write the Hamiltonian in a new form with center of mass coordinates Hˆ V M R r r which now allows us to separate the problem

18 Center of mass coordinates To separate the six-dimensional differential equation using these coordinates next, presume the wavefunction can be written Rr, SRUr Substituting this form in the Schrödinger equation with the Hamiltonian we obtain Hˆ M R r V r U S S M V U ES U R r r R R r r R r

19 Center of mass coordinates With U S S V r M U ES U R R R r r r R r then dividing by S R U r and moving some terms 1 1 RSR E V U S M U r r r R r The left hand side depends only on R and the right hand side depends only on r so both must equal a separation constant which we call E CoM E CoM

20 Center of mass coordinates Hence we have two separated equations M RS R E S R CoM Center of mass motion r V r U r EHU r Relative motion where E EE H CoM We can now solve these separately

21 Center of mass motion RSR ECoM SR M is the Schrödinger equation for a free particle of mass M with wavefunction solutions SR expik R and eigenenergies K ECoM M This is the motion of the entire hydrogen atom as a particle of mass M

22 Relative motion equation The other equation r V r U r EHU r corresponds to the internal relative motion of the electron and proton and will give us the internal states i.e., the orbitals and energies of the hydrogen atom

24 7.3 The hydrogen atom Slides: Video Informal solutions for the relative motion Text reference: Quantum Mechanics for Scientists and Engineers Section 10.3 ( Bohr radius and Rydberg energy )

25 The hydrogen atom Informal solution for the relative motion Quantum mechanics for scientists and engineers David Miller

26 Bohr radius and Rydberg energy We presume that the hydrogen atom will have some characteristic size which is called the Bohr radius a o We expect that the average potential energy strictly, its expectation value will therefore be e Epotential 4 a o o

27 Bohr radius and Rydberg energy For a reasonable smooth wavefunction r of size ~ a o the second spatial derivative will be 0 / o 0 / o ~ a a ao 0 / ao Note this is only meant to a rough estimate only within some moderate factor slope ~ 0 a o a o 0 a o ~ slope 0 a o

28 Bohr radius and Rydberg energy Remembering that for a mass the kinetic energy operator is / The average kinetic energy will therefore be Ekinetic ao Now, in the spirit of a variational calculation we adjust the parameter a o to get the lowest value of the total energy Such variational approaches can be justified rigorously as approximations for the lowest energy

29 Bohr radius and Rydberg energy With our very simple model, the total energy is e Etotal Ekinetic Epotential ao 4 oao The total energy is a balance between the potential energy which is made lower (more negative) by choosing a o smaller and the kinetic energy which is made lower (less positive) by making a o larger

30 Bohr radius and Rydberg energy For this simple model Etotal Ekinetic Epotential a o 4 o o differentiation shows that the choice of a o that minimizes the energy overall is 4 o a o 0.59 Å 5. e which is the standard definition of the Bohr radius We therefore see that the hydrogen atom is approximately 1 Å in diameter e 11 9 x 10 m a

31 Bohr radius and Rydberg energy With this choice of a o the corresponding total energy of the state is e Etotal ao 4o We can usefully define the Rydberg energy unit Ry e 13.6 ao 4o ev in which case Etotal Ry

32 Bohr radius and Rydberg energy Though we have produced the Bohr radius 4 o a o 0.59 Å 5. e 11 9 x 10 m e and the Rydberg Ry 13.6 ev ao 4o by informal arguments they will turn out to be rigorously meaningful The energy of the lowest hydrogen atom state is -Ry

8.1 The hydrogen atom solutions

8.1 The hydrogen atom solutions Slides: Video 8.1.1 Separating for the radial equation Text reference: Quantum Mechanics for Scientists and Engineers Section 10.4 (up to Solution of the hydrogen radial