1. Quantum Two Body Problem, Hydrogen Atom
Chapter 7
Quantum Two Body Problem,
Hydrogen Atom

2. 7.B.2
The two-body problem
The two-body problem: two spinless point objects in 3D interacting with each other (closed system)
Interaction between the objects depends only on the distance between them
The operators describing their positions and momenta satisfy these commutation relations:

3. The two-body problem The Hamiltonian of the system:
Let us introduce operators:
What commutation relations do these operators satisfy?

4. The two-body problem Let us calculate this commutator: Similarly:

5. 7.B.2
The two-body problem
On the other hand:
Similarly:

6. 7.B.2
The two-body problem
Thereby, after transformation we have operators of positions and momenta of two fictitious particles
How do the operators for real particles depend on the operators for fictitious particles? +

7. 7.B.2
The two-body problem
Thereby, after transformation we have operators of positions and momenta of two fictitious particles
How do the operators for real particles depend on the operators for fictitious particles? +

8. 7.B.2
The two-body problem
Let us now rewrite the Hamiltonian

9. 7.B.2
The two-body problem
Let us now rewrite the Hamiltonian

10. The two-body problem Let us now rewrite the Hamiltonian
M: total mass, μ: reduced mass
We converted our Hamiltonian into a sum of two separate Hamiltonians for two fictitious particles:

11. 7.B.2
The two-body problem
The two parts of the Hamiltonian commute with each other:
Therefore they both commute with the full Hamiltonian
Thus there should be basis common for all three operators:
In this case

12. The two-body problem In the coordinate representations:
The first equation is for a free particle and it is well known how to deal with it
The second equation is much more interesting from a physical viewpoint

13. 7.B.2
The two-body problem
This equation describes the behavior of two interacting particles in the center of mass frame and also the behavior of a single fictitious particle in a central potential
We can now drop the r subscript:

14. Particle in a central potential
The Laplacian in spherical coordinates:
Therefore:

15. Particle in a central potential
Since the orbital angular momentum operator depends only on the angular coordinates:
And:
Since H, L2 and Lz commute there is a common basis for all three of them

16. Particle in a central potential
Using the theory of the orbital angular momentum operator:
And:

17. Particle in a central potential
Using the theory of the orbital angular momentum operator:
And:
This equation and its solutions depend on the quantum number l as well as index k (that represents different eigenvalues for the same l) and does not depend on the quantum number m; thus:

18. Particle in a central potential
The behavior of the R functions at the origin should be sufficiently regular in order to represent a physical solution
The equation can be simplified via this substitution:

19. Particle in a central potential
The behavior of the R functions at the origin should be sufficiently regular in order to represent a physical solution
The equation can be simplified via this substitution:

20. Particle in a central potential
Therefore:
This function must be square-integrable:
Since the spherical harmonics are normalized
Quantum number l is called azimuthal, whereas quantum number m is called magnetic

21. 7.C.1
7.C.3
The hydrogen atom
A system of a proton and an electron can form a hydrogen atom
In this case the potential energy is:
And the reduced mass of the system:
Therefore, the radial eigenproblem becomes:

22. The hydrogen atom Let us make substitutions:
7.C.3
The hydrogen atom
Let us make substitutions:
This yields a dimensionless equation:
Therefore, the radial eigenproblem becomes:

23. The hydrogen atom Let us make substitutions:
7.C.3
The hydrogen atom
Let us make substitutions:
This yields a dimensionless equation:
What are the asymptotes of the solutions?

24. The hydrogen atom Let us make substitutions:
7.C.3
The hydrogen atom
Let us make substitutions:
This yields a dimensionless equation:
What are the asymptotes of the solutions?

25. 7.C.3
The hydrogen atom
Taking into account the asymptotes the solution could besought in the following form:

26. 7.C.3
The hydrogen atom
Taking into account the asymptotes the solution could besought in the following form:

27. 7.C.3
The hydrogen atom

28. 7.C.3
The hydrogen atom
Let us look for the solution in the following form of a polynomial:

29. 7.C.3
The hydrogen atom
Equating the coefficients of like powers of ρ yields:
The polynomial terminates at:

30. The hydrogen atom Therefore: Let us recall that:
Combining the two equations:
Defining:

31. The hydrogen atom Therefore: Let us recall that:
Combining the two equations:
Defining:
We obtain:

32. 7.C.3
Spectrum
Since
Conventionally and conveniently n is used to label the energy spectrum
n is called a principal quantum number
A given value of n characterizes an electron shell
Defining:
We obtain:

33. 7.C.3
Spectrum
Since
Conventionally and conveniently n is used to label the energy spectrum
n is called a principal quantum number
A given value of n characterizes an electron shell
Defining:
We obtain:

34. 7.C.3
Spectrum
Since
There is a finite number of values of l associated with the same value of n:
Each shell contains n sub-shells each corresponding to a given value of l
Defining:
We obtain:

35. 7.C.3
Spectrum
Since
There is a finite number of values of l associated with the same value of n:
Each shell contains n sub-shells each corresponding to a given value of l
Each sub-shell contains (2l + 1) distinct states associated with the different possible values of m for a fixed value of l

36. 7.C.3
Spectrum
The total degeneracy of the energy level with a value of En is:
Conventionally, different values of l are (spectroscopically) labelled as follows:
Subshell notations:

37. 7.C.3
Spectrum

38. 7.C.3
Spectrum

39. 7.C.3
Eigenfunctions
Let us synopsize all the transformations and assumptions for the eigenfunctions

40. Eigenfunctions Let us also recall the normalization conditions:
Now, using all this information, let us calculate the eignefucntions for the problem

41. 7.C.3
Eigenfunctions
We will start with the ground level, a nondegenerate 1s subshell
Normalizing:

42. Eigenfunctions We obtained the eigenfunction of the ground state!
It is completely spherically symmetric

43. 7.C.3
Eigenfunctions
What is the probability density of finding an electron in an elementary volume?
The probability of finding an electron between r and r + dr is proportional to
For the ground state this probability is thus proportional to

44. Eigenfunctions The maximum of this probability occurs at
Parameter a0 is known as the Bohr radius
Niels Henrik
David Bohr
(1885 – 1962)

45. 7.C.3
Eigenfunctions
The ground state function can be used to generate the rest of the eigenfunctions
E.g.,

46. Eigenfunctions The most general expression: Where
Is the associated Laguerre polynomial
Edmond Nicolas
Laguerre
(1834 – 1886)

47. Eigenfunctions Is the associated Laguerre polynomial 7.C.3
Edmond Nicolas
Laguerre
(1834 – 1886)

48. 7.C.3
Eigenfunctions
The radial parts of the eigenfunctions:

49. 7.C.3
Eigenfunctions
Probability density plots for the wave functions: