01-27-2008, 01:29 AM
Man - am I late to THIS party. FWIW, as a university physics instructor, I give TheTominator's response 8.5 for 10.
The problem isn't so much with TheTominator, as it is with the hand-waving way in which Ted King invoked the de Broglie wavelength.
These topics are not beyond a bright middle school student, but if you are going to talk about wave functions, then you have to talk about what they mean. Your student asked a very good question. A better question still is "Why do the electrons appear only occupy orbitals that correspond to whole number multiples of the de Broglie wavelength?"
Try this for a less nuance-dependent (but still hand waving) explanation:
As TheTominator points out, the de Broglie wavelength is inversely proportional to the linear momentum of the electron. The faster the electron goes, the shorter its "wavelength". This result is well supported by experiment, so you do not have to ask the students to accept it without proof.
As you go up the periodic chart, the charge on the nucleus increases -- which means that the electrostatic attraction between the nucleus and any electron in any orbit will increase.
It is a simple matter to demonstrate that the linear velocity of an object orbiting under the influence of an inverse square force (like the Coulomb interaction between point charges, or the gravitational attraction between widely separated masses, etc.) increases with increasing attraction.
That is to say, you could do just a little algebra on the board to show that, as the charge on the nucleus increases, the velocity (hence the momentum) of the electron will increase, meaning that its de Broglie wavelength will decrease. Smaller wavelengths mean a shorter resonant circumference for the mysterious standing wave rule of electron orbits.
As a bonus excercise, if you are comfortable with the math, you could work out the allowed orbitals for a planet in orbit around the Sun -- and show that the the quantum mechanical result duplicates the Newtonian result (and that, for a gravitational potential acting between the Sun and a planet, the difference between adjacent allowed orbital radii is vanishingly small)
edit: scary typo
The problem isn't so much with TheTominator, as it is with the hand-waving way in which Ted King invoked the de Broglie wavelength.
These topics are not beyond a bright middle school student, but if you are going to talk about wave functions, then you have to talk about what they mean. Your student asked a very good question. A better question still is "Why do the electrons appear only occupy orbitals that correspond to whole number multiples of the de Broglie wavelength?"
Try this for a less nuance-dependent (but still hand waving) explanation:
As TheTominator points out, the de Broglie wavelength is inversely proportional to the linear momentum of the electron. The faster the electron goes, the shorter its "wavelength". This result is well supported by experiment, so you do not have to ask the students to accept it without proof.
As you go up the periodic chart, the charge on the nucleus increases -- which means that the electrostatic attraction between the nucleus and any electron in any orbit will increase.
It is a simple matter to demonstrate that the linear velocity of an object orbiting under the influence of an inverse square force (like the Coulomb interaction between point charges, or the gravitational attraction between widely separated masses, etc.) increases with increasing attraction.
That is to say, you could do just a little algebra on the board to show that, as the charge on the nucleus increases, the velocity (hence the momentum) of the electron will increase, meaning that its de Broglie wavelength will decrease. Smaller wavelengths mean a shorter resonant circumference for the mysterious standing wave rule of electron orbits.
As a bonus excercise, if you are comfortable with the math, you could work out the allowed orbitals for a planet in orbit around the Sun -- and show that the the quantum mechanical result duplicates the Newtonian result (and that, for a gravitational potential acting between the Sun and a planet, the difference between adjacent allowed orbital radii is vanishingly small)
edit: scary typo