## Why Everything is Hard

This posting is a follow-up to my article from a few days ago asking, “Why is matter tangible?” At the end of that article, I derived the repulsive force that balances the attraction of the nuclei of metal atoms and resists penetration from outside— but only in metal. I could explain why a metal finger wouldn’t pass through metal, but I didn’t have a good argument for my non-metallic finger. There’s nothing like posting your ignorance to the world to rack your brain at night. Now I’ve got it: a general derivation of the contact force that applies to electron waves of any shape, under any potential. To my utter surprise, it is the Heisenberg uncertainty principle that makes everything hard!

The Heisenberg uncertainty principle, much like the Pauli exclusion principle, is a derived consequence of quantum mechanics, often mistaken for an axiom. Outside the context of quantum mechanics, it is called the Bandwidth Theorem, and is only a few steps removed from the Cauchy-Schwarz inequality. The most relevant special case for our purposes is $\Delta x \Delta p \ge \hbar/2$

which states that the uncertainty in a particle’s position times the uncertainty its momentum is at least $\hbar/2$, a non-zero constant. By “uncertainty,” I mean the width of the particle’s wave, which itself is a nebulous concept, because a wave can have any shape. We define it as the standard deviation of the particle’s waveform, treated as a probability distribution. $\displaystyle \Delta x = \sqrt{\int \psi^*(x)\psi(x) \, x^2 \, dx}$ and $\displaystyle \Delta p = \sqrt{\int \tilde{\psi}^*(p)\tilde{\psi}(p) \, p^2 \, dp}$

For simplicity, I chose the reference system that is centered on the particle, so that the average $x$ and the average $p$ are both zero (and the standard deviation is equal to to the root-mean-square, as above). $\psi(x)$ is the spacial amplitude (a function from space points to complex numbers), whose square is the probability density of the particle’s position, and $\tilde{\psi}(p)$ is its Fourier transform— amplitude as a function of momentum.

Uncertainty can be a misleading word, because it implies that the particle is at one of the $x$ points, and we just don’t know which one until we measure it. What’s really happening is that the particle starts in a state of single-valued energy and a continuum of space points— literally extended over a region of space. When we “measure” it, which means collide it with an energetic probe, the particle becomes single-valued in $x$ (the instant before it was scattered), and it chooses $x$ from the probability distribution $\psi^*(x)\psi(x)$. These states can be represented as (normalized) vectors in an infinite-dimensional vector space, in which there will always be some basis on which the state can be represented by a single axis, such as a single value of energy or a single value of $x$.

One of the great unsolved problems in quantum mechanics is, “How do particles in vivo choose their preferred basis?” When we look at electrons in the world outside, we usually find them in single-valued energy states if they’re bound to an atom, and mixed $x$/ $p$ states (such as a Gaussian wave packet) if they’re free. Atoms and larger structures, on the other hand, have firmly established $x$ positions, and can only be made to smear at low temperatures and in carefully-controlled environments. All I’ve been able to find that hints at an explanation is that electrons in bound states are constantly bombarded by very low-energy photons, much lower energy than the state they occupy, and that atoms in $x$ states are constantly bombarded by other atoms, with kinetic energies significantly above the energy of any spacially-smeared states they might otherwise form. I’d love to see a general explanation of this (and the whole process of decoherence), but for now we can take it as an empirical fact that electrons bound in solid matter are found in states of single-valued energy.

### Why Everything is Hard

In my previous article, I derived contact forces by observing the dependence between electron energy and volume. If we shrink the region of space in which the valence electrons are allowed to live, they fight back by demanding more energy. Force is defined by this relationship between energy and position. $\displaystyle F = -\frac{dE}{dx}$

To calculate the force, I used explicit knowledge of $E(x)$. How can we generalize the argument? The Heisenberg uncertainty principle provides a relationship between momentum and position, and momentum is not far removed from energy: $\displaystyle E = \frac{1}{2}mv^2 = \frac{p^2}{2m}$

This allows us to replace the variance of $p$ with the expectation value of $E$. $\displaystyle (\Delta p)^2 = \int \tilde{\psi}^*(p)\tilde{\psi}(p) \, p^2 \, dp = 2m \int \tilde{\psi}^*(p)\tilde{\psi}(p) \, E \, dp$

Since we’re talking about electrons in matter, the bound states are going to be single-valued in energy, so the expectation value is simply the unique energy of the state. Now Heisenberg’s uncertainty principle becomes $\displaystyle E \ge \frac{\hbar^2}{8m(\Delta x)^2}$

For any system of electrons in matter, the energy will diverge as the size of the state, $\Delta x$, goes to zero. Thus, if I press my finger to a block of wood, the enormously complicated system which consists of the electrons in my finger and the electrons in the wood resist the shrinking of available electron space that would occur if my finger passed into the wood. I’m not strong enough to lift Avogadro’s number of electrons into excited states, nor is the attraction in the wood strong enough to keep it from breaking apart if I tried to do that. Instead, the whole block slides away from the onslaught, or it crumbles and squishes around my finger, or it compresses a little bit (especially if it has air pockets to collapse). Figure 1. Two objects side-by-side: all electrons are at the lowest energy states available to them (except for thermal fluctuations). Figure 2. If two objects share the same space, the total volume available to electrons is reduced. The energy level of many of the electrons involved would be significantly increased. Figure 3. What really happens: it takes much less energy to twist the book, because that leaves the electron-volume nearly the same— it strains electrostatic attraction of neighboring molecules instead, a more cost-effective solution.

(This is not the paperback edition.)

There are sequences of energy states that run counter to the rule that smaller wave size means higher energy. For instance, electrons bound to a simple atom decrease in energy as they decrease in size. But these sequences have a minimum $\Delta x$ in the ground state: no lower-energy states are possible, so we don’t get to test the $\Delta x \to 0$ limit. Because the uncertainty principle is an inequality, it only says that energy will eventually diverge when $\Delta x$ is small enough. To be precise, it’s not saying that everything is hard, but rather that nothing is infinitely soft.

The dependence of energy on available space doesn’t even need to be monotonic. We could imagine a material which, when partially squished, collapses of its own accord, forming a harder, denser substance. Is coal/diamond an example?

The most surprising thing to me is that this relationship between energy and size is not limited to electrons, and not even to spinor-particles like the Pauli exclusion principle. Any particle will resist being crammed into a small space, even photons. In fact, photons in the cosmic microwave background are losing energy due to this effect right now because space is getting bigger!

So does this mean that neither Explanation #1 (electromagnetism), nor Explanation #2 (the Pauli exclusion principle) is the real reason that stuff is hard? It’s certainly more complicated than either one in isolation.

1. Electromagnetism is the origin of the potential that confines electrons to nuclei, the inertial mass that we call matter.
2. The Heisenberg uncertainty principle ensures that, at some “small enough” scale, energy will increase as the available electron space shrinks. This is the contact force.
3. The Pauli exclusion principle can multiply this force by a huge factor ( $10^{15}$ in the case of metal) because without it, all the electrons would sit in the ground state. This is what makes stuff harder.

Quantum mechanics has certainly complicated the picture. We used to think that matter is hard because, well, it’s matter, being hard is what makes it matter. Then we had to replace that picture to properly describe wave-particles, including the fact that sometimes wave-particles pass right through each other (and us) like ghosts. Then we had to invoke one Basic Interaction and two Principles to make ordinary matter hard again. But look what we’ve done: instead of just describing how matter moves, as in classical physics, we’ve begun to describe what matter is!

### 3 Responses to “Why Everything is Hard”

1. Aaron F. Says:

Oooooooh, cool! It’s great to finally hear a solid explanation for something that everyone traditionally glosses over. Incidentally, another such frequently-glossed concept is the uncertainty principle itself… I’ve never seen a decent derivation of it, and now that you’ve mentioned the Cauchy-Schwarz inequality, I’m tempted to have a go at it myself. 🙂

p.s. I’m still having trouble with the idea that the electrons in your finger and the book would have less available living space if the two objects interpenetrated. Consider two electrons in two potential wells of width L. If the wells are separate, each electron has a position uncertainty of ~L. If the wells overlap, both electrons still have a position uncertainty of ~L! To me, therefore, it makes more sense to say “It’s hard to make the wells overlap because one electron must gain energy due to the Pauli exclusion principle” than to say “It’s hard to make the wells overlap because both electrons gain energy when their living spaces are reduced.” Of course, I’m not the world’s brightest quantum mechanic… 😛

2. jpivarski Says:

The uncertainty principle actually comes from two inequalities. The Cauchy-Schwartz inequality is the first one; the second is the observation that |A|^2 >= Im{A}^2. Wikipedia has an okay derivation (step-by-step, but with little explanation), but the best derivation I’ve seen is in Shankar’s red textbook.

After posting this, I wished I had written the reduced-space argument more carefully. The Figures were fun to make, but they don’t add much. Here’s what I’m thinking: when two wells of length L are not overlapping, the energy of electrons in those wells is independent of separation, in agreement with our observation of zero contact force between objects which are not in contact. When the wells just barely touch each other, the length is 2L. As they overlap more, the total length shrinks to less than 2L, reaching a lower limit of 1L when they completely overlap. The force I derived is the force needed to shrink the well, starting at 2L with all electrons filling the entire combined well.

I never handled the issue of what happens when separated blocks *first* come into contact. For infinite square wells, there’s a net reduction in potential energy:

energy of separated metal blocks > energy of combined block

2 Sum_n^N (n/L)^2 > Sum_n^2N (n/2L)^2

but this difference is tiny in the limit of large N. This means there’s a tiny attractive force between metal blocks until they touch and conduct electrons: an attractive contact force! It would be hard to measure such a force in a macroscopic block because there are a lot of other small attractive forces (polarization, the Casmir effect, residual charge, and so on).

(For two blocks with 1e23 valence electrons, the fractional energy loss is 0.75e-23. Multiply this by the number of electrons, and we’ve only lost a quarter of an average electron’s energy, for the whole block. It might be noticible in nanofabricated systems.)

3. alex Says:

Everything is hard because energy in itself is indestructible and ultimately unlimited. Everything is made of energy and is the building blocks of our collective consciousness. When you try hard and feel the weight, its because your energy is colliding with the surrounding energy generating more energy into existence creating a pressure. Everything is hard because it exists, the harder it is, the more energy inside. Pressure increases with energy, which is why we must balance ourselves and our environments-peace-AG