What would you consider to be the most quintessential equations of QM?

[SnotRocketSci]

I am an UG student majoring in Physics. I am just starting classical coursework, but I have had, for the past 10 or so years, a great interest in Quantum Theory. I have read nearly every popular-level (read: very little, if any, math) book published in the last 15 years on the subject, but just over the last 2 years have I really started to study the prerequisite math to really get into it.

So, I am actually doing an art project that will have QM as it's theme; the background of which will be equations. So I guess what I am asking is what do you believe are the most important equations, expressions, etc in QM that really encapsulate the theory?

So, obviously, Heisenberg's Uncertainty inequality; Schrödinger's Equation (but which form of it). What else? I need 5 in total.

Thanks in advance for your opinion/help!

 

[Simon Bridge]

Schrödinger Equation:$$-i\hbar \frac{\partial}{\partial t}\Psi = H\Psi$$

However - it is quantum interference that really spells out the departure from classical mechanics ... see: http://arxiv.org/pdf/quant-ph/0703126] for the full quantum wave-mechanics formalism.

 

[kith]

If I had to choose five equations, I would use the Schrödinger equation, the probability for measuring a given eigenvalue, the HUP, the symmetry of a state of identical particles and the Dirac equation.

More buzzwords: Schrödinger's cat state, Bell's inequality

For elegance, I would use the abstract bra-ket formalism. If the art is directed at people with a little QM background I would maybe use the wave mechanics form of the equations to connect with their previous knowledge.

 

[mfb]

The Standard Model lagrangian. Some of its shorter versions, like that.
It is quantum field theory and not nonrelativistic quantum mechanics, but if you are looking for "one equation to describe physics", this is probably the closest you can get (with current knowledge of physics). If you need 5 equations, I think this is a good one to include.
+ Schrödinger equation in the way Simon Bridge posted it
+ Uncertainty principle as position/momentum ($\sigma_x \sigma_p \geq \frac{\hbar}{2}$) or in its general version $\sigma_A \sigma_B \ge \frac{1}{2}\left|\langle\psi|[\hat{A},\hat{B}]|\psi\rangle\right|$
+ the de-Broglie wavelength $\lambda = \frac{h}{p}$
+ Dirac equation

energy levels in an hydrogen atom? $E_n = \frac{-13.6 eV}{n^2}$
maybe the expectation value of an operator? $\langle A \rangle = \langle\psi|A|\psi\rangle$

Something related to the double slit would be nice, but I don't see how this can be compressed to a single equation without an additional sketch.

 

[SnotRocketSci]

Could someone please write out the *Time-Dependent* version of the Schrödinger Equation for me?

I have seen it written a few different way with there being subtle changes in the way the variables, operators were arranged.

 

[SnotRocketSci]

+ Uncertainty principle in its general version $\sigma_A \sigma_B \ge \frac{1}{2}\left|\langle\psi|[\hat{A},\hat{B}]|\psi\rangle\right|$

Could you possibly talk me through this version of it?

 

[Simon Bridge]

Could someone please write out the *Time-Dependent* version of the Schrödinger Equation for me?

That was what I wrote out.
http://en.wikipedia.org/wiki/Schrödinger_equation#Time-dependent_equation
Oh hell I put a minus sign in there... that would make the Hamiltonian negative...
All the other versions are algebraically equivalent to this one - there are lots of ways of expressing things and you can even pick your units so the $\hbar$ = 1.

 

[SnotRocketSci]

That was what I wrote out.
http://en.wikipedia.org/wiki/Schrödinger_equation#Time-dependent_equation
Oh hell I put a minus sign in there... that would make the Hamiltonian negative...
All the other versions are algebraically equivalent to this one - there are lots of ways of expressing things and you can even pick your units so the $\hbar$ = 1.

So the second equation (in the white box) down is equivalent to the way you wrote it above?

And thank you for your help! The audience viewing this does know their stuff, and while they would totally understand that I made a mistake because I am probably 2 years away from this level of mathematical knowledge, I don't want to have a work that I will have 20 hours + into be wrong, haha. So thanks!

 

[Simon Bridge]

So the second equation (in the white box) down is equivalent to the way you wrote it above?

I've just got a stray minus sign at the front that should not really be there. I'm also used to leaving the little hat off the operator symols - it's a common shorthand.

The second box down in wikipedia is the same as the first box when the hamiltonian operator is given by: $$ \hat{H}=\frac{-\hbar^2}{2m}\nabla^2 + V(\vec{r},t)$$

So I should have written: $$i\hbar\frac{\partial}{\partial t}\Psi(\vec{r},t)=\hat{H}\Psi(\vec{r},t)$$

 

[mfb]

$\sigma_A \sigma_B \ge \frac{1}{2}\left|\langle\psi|[\hat{A},\hat{B}]|\psi\rangle\right|$

Could you possibly talk me through this version of it?

A and B are arbitrary operators. [A,B] is the commutator: [A,B]=AB-BA. While this looks like 0, operators do not always commutate in quantum mechanics: AB does not have to be the same as BA.
With position and momentum as example, the left-hand side becomes $\sigma_x \sigma_p$. Position and momentum operator do not commute in quantum mechanics, and $[\hat{x},\hat{p}]=\hat{x}\hat{p}-\hat{p}\hat{x}=i\hbar$. In that way, you get the momentum/position uncertainty relation from the general uncertainty relation.

In the momentum basis, where $\hat{x}=x$ and $\hat{p}=-i\hbar \nabla$:
$\langle\psi|[\hat{x},\hat{p}]|\psi\rangle = -i \hbar \langle\psi|x\nabla - \nabla x|\psi\rangle = -i \hbar\left( \langle\psi|x\nabla|\psi\rangle - \langle\psi|\nabla x|\psi\rangle\right) = -i \hbar\left( \langle\psi|x\nabla|\psi\rangle - \langle\psi|x\nabla|\psi\rangle - \langle\psi|\psi\rangle\right) = i \hbar$ where the third = uses the product rule