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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# The Real Hydrogen Atom

### Why do this problem?

This problem provides a glimpse into the way differential equations
develop at university both in terms of mathematical structure and
also in their applications. It is ideal for high achieving maths
students coming to the end of their school careers who wish to draw
their learning together or to prepare themselves for the university
challenges ahead.
### Possible approach

### Key questions

### Possible extension

### Possible support

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### Differential Equation Matcher

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Links to the University of Cambridge website
Links to the NRICH website Home page

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30 April (Primary), 1 May (Secondary)

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This problem is most suitable for the highest attaining
mathematics students or those with a very strong interest in
pursuing mathematical physics.

One thing might strike students: the problem involves a lot of
reading and several parts to consider and solve. Encourage those
working on it to read each part carefully and consider the
mathematical ideas raised: the thinking about the issues can be of
as much importance as doing the problem; reflecting on the solution
obtained rather than simply getting the algebra correct is crucial!
Furthermore, students who have become accustomed to completely
solving problems might need to be reminded that partial solutions
are acceptable and that at higher educational levels complete
solutions and not the norm.

It is often a good idea to encourage students to talk about
their mathematical experiences. Perhaps this problem could be given
to students who consider the problem with plenty of thinking space
and then give a short presentation on the problem to the rest of
the group at a later date. Alternatively, small groups of
interested students could consider the problem collectively and try
to make sense of the ideas, being proactive in helping themselves
out of difficulty.

The problem is well structured into different parts, so could
be attempted by students independently. Some good questions to help
students out of difficulty are:

Are you clear as to what is constant and what is variable in
each case?

Have you understood the meaning of the coordinate
system?

Have you understood the meaning of the variables?

Do you have a plan: do you know what you are trying to solve?
What sort of maths might help you to solve it?

There is plenty of extension built into this problem.
Interested students might also try more of the problems from the Advanced Scientific
Mathematics section of stemNRICH.

There are many more straightforward differential equation
problems on the Advanced Scientific
Mathematics section of stemNRICH.

Match the descriptions of physical processes to these differential equations.

Things are roughened up and friction is now added to the approximate simple pendulum

As a capacitor discharges, its charge changes continuously. Find the differential equation governing this variation.