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When applied to the hydrogen atom, the wave function should describe the behaviour of
both the nucleus and the electron,
.
This means we have a two body problem, which is very difficult to solve.
We can fortunately convert this two-body problem to an effective one-body
problem by transforming from the *Laboratory System* to the
*Centre of Mass System*.

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The appropriate potential is of course the simple radial electrostatic potential
for a point charge nucleus of charge . (Note that if we are not dealing with hydrogen, then we are dealing with hydrogen-like atoms which are fully stripped of all but one electron.)

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**Exercise 1**
Writing the double differential operator in spherical polar co-ordinates is not trivial.
Check *Electromagnetic Fields and Waves* by Lorrain and Carson, Chapter 1, if you require
insight into the last step. You are not expected to be able to do this transformation.
However, make sure, using a sketch, that you can show how an infinitesimal volume element
behaves under this transformation.

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**Exercise 2**

Write each of the variables
in terms of the variables ,
also perform the inverse mapping. You may use figure 1.
Note that the potential is radial,
, which means
it depends only on , and not on or .

The wave function necessarily is separable into radial, polar and azimuthal factors
under a radial potential as follows :
.
(Once again, can you think why ?)
Substituting the above expression and the potential into the spherical polar
representation of the wave equation, we find, after some manipulation,

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**Exercise 3**

Ensure you can achieve the last result with your own pencil and paper. Doing these exercises and
tutorials properly helps to make you familiar
with quantum mechanics, via your fingertips, into the marrow of your bones. You may use your textbooks
the first time you do this.

This equation has on the left functions of only, and on the right a function of
only. Accordingly, it can only be satisfied for all values of the independent variables by requiring
that both sides are equal to the same constant value. Hence it follows the the left hand side, which is
called the azimuthal equation, is equal to a constant, which is called the azimuthal constant.
We give the azimuthal constant the symbol , for reasons which will become clear with
hindsight.

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The solution for the azimuthal equation (by the ansatz method) is

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**Exercise 4**

Check this with your own pencil and paper....

Imposing the boundary conditions, we must have single valued, and
. Therefore, we require the constant to be quantised
as follows.

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We call the *magnetic quantum number*. You may imagine that the quantity
measures the change of the azimuthal part of the wave function
with change of the azimuthal co-ordinate. Thus this quantity can sense
differences under a rotation about the -axis. Intuitively, if the wave function of the
electron changed under such a rotation, one would be able to discern it, and classically,
a rotating charge has a magnetic moment. This is a heuristic justification of the appellation
of ``magnetic quantum number'' for .

Clearly, not only is the separation constant quantised, but also the azimuthal wave function becomes
part of a family of wave functions labelled by the quantum number .

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Congratulations !

You have just seen that the quantisation of the magnetic quantum number arises naturally from the condition that the wave function must be single valued and satisfy its boundary conditions. The imposition of boundary conditions also lead to quantisation of the wave function in the previous examples we have seen (particle in a box). This is just a mathematical way of saying we have successfully trapped a wave function in an attractive potential. The value of the prefactor will be set later via a

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We have achieved so far separated equations for the last two wave functions, viz. the
radial wave equation for and the polar wave equation for
. The solution of these two equations is beyond the scope of this course.
Rest assured, it proceeds as in the case for the azimuthal wave function. That is, imposing
the boundary conditions causes the separation constant to become quantised and also the
radial wave function and the polar wave function to become part of a family labelled by the
appropriate quantum number.

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(The angle dependent wave functions are known as the Spherical Harmonic Functions, and the radial wave functions as Laguerre polynomials. These solutions are tabulated below in figure 2.)

Hence we find, on solving the
polar wave equation for
that

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**Exercise 5**

What do you think the boundary conditions are for the polar wave equation ?

In the introductory course on Quantum Mechanics, we saw that was also a separation constant.
It arose when we separated the time and space parts of the *Time dependent wave equation*
to arrive at the *Time independent wave equation*, which we have presented at the top of this section.
So it will come as no surprise now to find that becomes quantised on requiring that
the solutions of the radial wave equation above also obey boundary conditions. In the case of the radial
wave equation, we obviously require that is finite and
.
We get ,

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**Exercise 6**

This is the same expression as in the Bohr model.
Muse on what this correspondence with the Bohr model implies.

**Exercise 7**

How would these results change if we were dealing with positronium and not hydrogen ?

We may summarise our results so far :

The Schrödinger Wave Equation for hydrogen like atoms in three dimensions is best treated in spherical
polar co-ordinates

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In this process, we also find a family of wave functions, labelled by the quantum numbers :

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*Observation*

Schrödinger sure deserved the Nobel Prize !

**Example**

Suppose we want to verify the energy of the ground state wave function of the
hydrogen atom, and
. We note that it is only the radial wave
equation which contains , the energy of the state. The appropriate radial wave function
is

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