On the Importance of Trig Identities in Quantum Mechanics

Just a few random thoughts on why I wish I'd done a better job of memorizing my trig identities in high school.

Here's an approximate high school conversation of mine regarding trigonometric identities with my teacher Mr. Tully, (who is awesome and by the way, who is also a rancher, and also by the way is not the guy pictured to the left(picture 1)... read on!)

Me: "Why do I have to memorize these 40 or so trig identities[2]?" (yes even then, I referenced my utterances"

Mr. Tully: "Because they'll be very important for what you want to do later." (he knew I wanted to be a physicist)

Me, (typically not thinking 'later' might be after next week): "Yeah... I'm not seeing it..."

Fast forward a bit to grad school. Twice in the last week, trig identities were make or break features of homework problems. I didn't pick up on the necessary identities in electromagnetism, and I'll probably get a B instead of an A on my homework. A B in grad school is a C in undergrad. Even more importantly, I didn't get to see the 'cool features', (no, really, I'm sure they were cool features), of the solution because I didn't get to it.

In quantum mechanics homework tonight, I was working on a tunneling problem. I used the following two identities and wouldn't have been able to move forward without them (picture 2):

What's tunneling? In the world we're used to, when an object encounters a barrier like the one shown below(picture 3), it will just bounce off.

In the energy and size levels where quantum mechanics applies though, a particle can actually just move right through the barrier and turn up on the other side. So, why's that important?

Here's one reason. If you have an understanding of quantum tunneling, then you can do things like the guy pictured above (picture 1), Ivar Giaver, who before he even had a PhD performed an experiment that found that electrons could tunnel from a superconductor across an insulating barrier into another superconductor. After that, he won the Nobel prize. His discovery also enabled magnetic resonance imaging, (MRI), technology to be used as a medical diagnosis tool, which is kind of cool.

And that is why I should have memorized my trig identities.

References:
1. http://en.wikipedia.org/wiki/Ivar_Giaever

2. http://en.wikipedia.org/wiki/Trigonometric_identity

Picture of the Day: (picture 4)

From 2/7/13

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