Stepping into the quantum world

Part 1 of the quest for the hydrogen molecule.

There is no way around it: we are going to do quantum mechanics. So let’s start with an introduction to establish a starting point and get ourselves familiarized with the general ideas. After that, we are going to start calculating!

There are several ways you can start building a theory of quantum mechanics. Basically these are the main options:

  1. Schrödinger equation
  2. Hilbert space states
  3. Path integrals

This is also the order by which they were discovered, and of course it also became the order in which quantum mechanics is usually taught. Following the exact footsteps of previous physicists is a great way not to make progress, since you will build the same concepts and misunderstandings in your mind as the people before you. We will try to choose our own path, where possible.

So, let’s compare them.

  1. The normal procedure would be to start with the Schröding equation and then go calculate stuff. Which is fine, but it is also not very illuminating. Where does this equation come from? What does it tell us about the world, really? It does have the advantage that you can actually calculate things, so we will have to use it at some point. It is not much use when you go to more advanced topics  (relativistic quantum mechanics, quantum field theory).
  2. Hilbert space states are nice but a a little abstract in the beginning. They are good to develop first ideas and make you feel closer to “the truth”.
  3. Path integrals are very cool, and quite useful for keeping things simple (relatively speaking) when you go to more advanced physics. They are mathematically fuzzy and strange. They are not simple to calculate and a bit frightening. We can take a look later if it is possible to calculate the hydrogen atom with path integrals, because I don’t know really.

So let’s start with the second idea, and focus on quantum states. We will learn to reason from the abstract to the concrete and we will also learn that the world is a bit more… orthogonal (?) than what we think it is.

A state is a description of a situation. We will use this notation to write down a state: \(|\rm{state}\rangle\). Furthermore the states can be added and subtracted, and they can be multiplied by a number. Don’t worry if this feels a little abstract, it’s supposed to be that way. We don’t make any assumption about the underlying structure of such a state.

Let’s start with an example. If I am feeling happy, I could describe my state as

\(|\rm{happy}\rangle\).

Since states can be multiplied with -1, this is another state:

\(-|\rm{happy}\rangle\).

Question: which of these states is happier? Answer: they are both equally happy. An unhappy state would be an entirely different one:

\(|\rm{unhappy}\rangle\).

What if I am both happy and tired, how can I describe that? We could write it down like this:

\(|\psi\rangle=|\rm{happy}\rangle+|\rm{tired}\rangle\),

where I have given the combined state a name “\(|\psi\rangle\)“. The simple rule is: all states can be added. So that means we can also have the state:

\(|\psi\rangle=|\rm{happy}\rangle+|\rm{unhappy}\rangle\).

Really? It looks confusing, so we must be doing quantum mechanics!

Next time, we are going to derive the Schröding equation, although it is generally agreed that this is not possible.

The quest to understand the hydrogen molecule

I admit that I don’t really understand the hydrogen molecule.

It is the simplest molecule but it seems so complicated to have a feeling for how it really works, or to explain it well. Wouldn’t it be great to understand it thoroughly? And with it all the physical laws that make it the way it is? It would give a great view on the way things really are, to grasp the underlying simplicity that is so strange to us.

Perhaps we should go on a quest. A journey with many steps, and at each one we learn something that is real and that is true. It will be a difficult quest. We may not even make it. Not all of us will make it. And we may take wrong turns or have to come back to a point where we were before. We may take more difficult steps where we could have taken a simpler path. Who knows where we will end up. But it will be quite educational. If you can follow the path. We’ll see.

Save money by driving

Do you enjoy driving? Do you care about pollution? Do you like money? If your answer to one of these questions is yes, the following may be of interest to you.

Last summer it was quiet on the road during my daily commute. This enabled me to do a little experiment that I have been wanting to do. I know my car uses less fuel when driving slower, but how much less?

To test this, every day I would drive 100 km/h on the highway in the morning and 120 km/h in the evening when coming back, or vice versa, chosen more or less randomly. Every time I recorded fuel consumption for my Fiat 500, travel time, and temperature.

This gave me some data points, that I could analyze. Now to be fair, I have only 13 data points, so it is not all very scientific. But citizen science is mostly about the fun, right?

Here are the data points I recorded, plotted against temperature:

Fuel consumption versus temperature

Now be careful, the fuel consumption is for the whole trip, while the change in driving speed was only applied to the highway part. Let’s see if the trip time correlates with the speed.

Time vs inverse speed

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)     21.143      3.391   6.235 6.39e-05 ***
inverse_speed   40.714      6.098   6.677 3.48e-05 ***

Residual standard error: 1.096 on 11 degrees of freedom
Multiple R-squared:  0.8021,    Adjusted R-squared:  0.7841 
F-statistic: 44.58 on 1 and 11 DF,  p-value: 3.483e-05

(sorry for being confusing with the inverse speed, but it is the physically correct quantity to use). That correlation is certainly significant, but there is some variation. Probably this is due to random events, like traffic lights or small traffic jams. The correlation is 40.714 km, meaning that effectively 40.7 km of the 60 km drive was on highways.

The question rises whether it is better to use the correlation of fuel consumption with speed, or with travel time. Because of the variations mentioned above, I actually find a more significant correlation with speed. I will use this to calculate how much money you save when driving slower.

Here are the correlations with all parameters that seem relevant:

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  2.055750   0.655437   3.136 0.011996 *  
speed        0.028449   0.004505   6.316 0.000138 ***
temperature -0.028659   0.027249  -1.052 0.320333    
direction    0.290119   0.144276   2.011 0.075223 .  

Residual standard error: 0.1496 on 9 degrees of freedom
Multiple R-squared:  0.8734,    Adjusted R-squared:  0.8312 
F-statistic:  20.7 on 3 and 9 DF,  p-value: 0.0002236

All correlations for fuel consumption

Indeed, the correlation with speed is very significant. There is also a correlation with direction (morning or afternoon drive), probably due to typical wind conditions. The correlation with temperature is not very significant, but in my experience it really becomes significant when you compare summer and winter driving conditions.

The speed / fuel consumption coefficient is 0.028449, or, in other words, by driving 20 km/h slower I save 0.56898 liter per 100 km. But I assumed that the savings were only due to the highway part of our trip. So I should calculate the absolute numbers: the full trip is 59.9 km, so the saving is 0.3408 liter. For just the highway part, this is 0.8374 liter per 100 km.

The average price of Euro95 right now is 1.600 euro/liter. This means that I save 1.34 euro per 100 km. When driving 120 km/h, driving 100 km takes 50 minutes, but at 100 km/h, it takes 60 minutes, a difference of 10 minutes.

Of course, I assume here that all the correlations are linear, but I know that they are not. In fact, if you drive faster, say 130 km/h or 140 km/h, your fuel consumption will increase more dramatically, and the savings will be bigger.

Conclusion

By driving slower on the highway, you can save 1.34 euro, but you arrive 10 minutes later. This means you save 8 euro by driving for one hour. Maybe not enough to earn a living, but if you have the time: relax and drive slower!

If you are interested, you can get the  data and the R script that I used.