Where do the shapes of the orbitals come from? Where is the electron in the p orbital? High school students often have a hard time reconciling the tangible idea of an electron with the idea that the electron has wave-like properties. This is understandable as they move from the Bohr model with electrons moving up and down energy levels as discrete particles to occupying “electron clouds” that have specific shapes. A student may wonder how does the idea of electrons with wave-like properties relate to the electron clouds?

Some textbooks avoid the conceptual conflict between particles and waves by dropping the idea of referring to wave-like properties of electrons at all. They refer to the shape of the orbitals coming from the probability of finding an electron, and a sack around the cloud where the electron has a 90% chance of being found defines the shape of the sack. There is no reference to waves or the “wave-mechanical model”.(Matta et al. 2012)

Other texts use the idea of waves. They describe the quantum mechanical model as being also called the wave mechanical model, and that Schrodinger treated hydrogen’s electron as a wave in his equation. They also go on to describe that solutions to the equation are called wave functions, and that these wave functions predict a three dimensional region around the nucleus called an atomic orbital. (Buthelezi et al. 2009)

Having some concrete analogues often helps in understanding any topic. Concrete analogues for quantum mechanics include dice throwing (Neto 1984), music (Eagle, Seaney, and Grubb 2017) and die itself (Fleming 2001) and a concrete analogue may help here, even it is only two dimensional. I would like to suggest such a concrete analogue I have used in my classroom.

With a student volunteer in front of the class. I take a long-coiled spring capable of stretching out to about 3 meters, and we gently move it up and down, so we have a repeating half wavelength. I note that this repeating pattern is called a standing wave - it is not going anywhere. I then ask students if we could have a lower energy standing wave? They will tell me no. Considering both halves of the wave, up and down, I ask them what orbital does this resemble? They will tell me the s orbital. I then ask my student to help make the next highest standing wave which we do by putting noticeably more energy into the slinky. We end up with a full wave moving up and down. Considering the full range of motion of the coiled spring, the p orbital becomes visible.

This simple demonstration shows a plausible connection between electrons as waves and the shapes of the s and p orbitals. It may encourage an understanding that goes beyond memorizing unrelated facts.

- Buthelezi, Thandi, Laurel Dingrando, Nicholas Hainen, Cheryl Wistrom, and Dinah Zike. 2009. New York Chemistry Matter and Change. Columbus, Ohio: McGraw-HIll.
- Eagle, Forrest W., Kyser D. Seaney, and Michael P. Grubb. 2017. "Musical Example To Visualize Abstract Quantum Mechanical Ideas." Journal of Chemical Education 94 (12): 1989-1994. https://doi.org/10.1021/acs.jchemed.7b00413.
- Fleming, Patrick E. 2001. "A Quantum Mechanical Game of Craps: Teaching the Superposition Principle Using a Familiar Classical Analog to a Quantum Mechanical System." Journal of Chemical Education 78 (1): 57. https://doi.org/10.1021/ed078p57.
- Matta, Michael, Dennis Staley, Antony Wilbraham, and Edward Waterman. 2012. Chemistry. Upper Saddle River, New Jersey Pearson.
- Neto, Benicio de Barros. 1984. "Dice throwing as an analogy for teaching quantum mechanics." Journal of Chemical Education 61 (12): 1044. https://doi.org/10.1021/ed061p1044.

## NGSS

Students who demonstrate understanding can evaluate the claims, evidence, and reasoning behind the idea that electromagnetic radiation can be described either by a wave model or a particle model, and that for some situations one model is more useful than the other.

*More information about all DCI for HS-PS4 can be found at https://www.nextgenscience.org/topic-arrangement/hswaves-and-electromagnetic-radiation.

Students who demonstrate understanding can evaluate the claims, evidence, and reasoning behind the idea that electromagnetic radiation can be described either by a wave model or a particle model, and that for some situations one model is more useful than the other.

Assessment does not include using quantum theory.

Emphasis is on how the experimental evidence supports the claim and how a theory is generally modified in light of new evidence. Examples of a phenomenon could include resonance, interference, diffraction, and photoelectric effect.