that is uncertain, but limited to values that keep it near the nucleus. Bohr’s idea of allowed energy states still applies—the electron will always have one of the limited number of energy values predicted by Bohr’s theory—but these states no longer correspond to electrons moving in particular orbits.
“So, wait—the electron isn’t in a particular place, it’s just kind of near the atom?”
“That’s right. The different energy states correspond to different probabilities of finding the electrons at particular positions, and higher-energy states will give you a better chance of finding the electron farther from the nucleus than lower-energy states. But for any of the allowed states, the electron could be at just about any point within a few nanometers of the nucleus.”
“But what happens if you have two atoms close together?”
“Well, if you bring two atoms close enough together, an electron that starts out attached to one atom can end up on the other atom, because of this quantum uncertainty in the position. We’ll talk a little more about this in chapter 6 , when we talk about tunneling.”
“Okay.”
“You can also get situations where an electron is sort of ‘shared’ between two atoms. That’s how chemical bonds form. And if you get a bunch of atoms together in a solid, one electron can be shared among the whole solid. That’s the basis for the quantum theory of solids, which lets us understand how metals conduct electricity and how to make semiconductor computer chips. It’s all because electrons extend beyond specific planetary orbits.”
“Uncertain electrons are weird.”
“Strictly speaking, it’s not just electrons. Everything in the universe is subject to the uncertainty principle, and has an uncertain position and velocity.”
“That can’t be right. I mean, I can see my bone right over there, and it has a definite position, and a velocity of zero.”
“Ah, but the
quantum
uncertainty associated with your bone is dwarfed by the
practical
uncertainty involved in measuring it. If you look at it really carefully, you might be able to specify its position to within a millimeter or so—”
“I
always
look at my bone carefully.”
“—and with heroic effort, you might bring that down to a hundred nanometers. In that case, the velocity uncertainty of your hundred-gram bone would be only 10 -27 m/s. So, the velocity would be zero, plus or minus 10 -27 m/s.”
“That’s pretty slow.”
“Yeah, you could say that. At that speed, it would take the age of the universe to cross the thickness of a single atom.”
“Okay, that’s
really
slow.”
“We don’t see quantum uncertainty associated with everyday objects because they’re just too big. We only see uncertainty directly when we look at very small particles confined to very small spaces.”
“Like electrons near atoms!”
“Exactly.”
Uncertainty has another, even more profound effect on the structure of atoms. Electrons must always have uncertainty in both their position and momentum, and that means that the energy of an electron in an atom can never be zero. To have zero energy while still being part of an atom, an electron would need to be not moving, sitting right on top of the nucleus. This is impossible, as we’ve already seen—the closest we can come is to make a narrow electron wave packet centered on the nucleus, which will include lots of different states with nonzero momentum.Even the lowest-energy-allowed state of hydrogen, then, has some energy.
This is a general phenomenon, and applies to any confined quantum particle. If we know that a particle is in some particular region of space, that limits the uncertainty in the position, and increases the uncertainty in the momentum. Confined quantum particles are never at rest—they’re like puppies in a basket, always squirming and wiggling and shifting around, even when they’re asleep.
This tiny residual motion is called zero-point energy,
“So, wait—the electron isn’t in a particular place, it’s just kind of near the atom?”
“That’s right. The different energy states correspond to different probabilities of finding the electrons at particular positions, and higher-energy states will give you a better chance of finding the electron farther from the nucleus than lower-energy states. But for any of the allowed states, the electron could be at just about any point within a few nanometers of the nucleus.”
“But what happens if you have two atoms close together?”
“Well, if you bring two atoms close enough together, an electron that starts out attached to one atom can end up on the other atom, because of this quantum uncertainty in the position. We’ll talk a little more about this in chapter 6 , when we talk about tunneling.”
“Okay.”
“You can also get situations where an electron is sort of ‘shared’ between two atoms. That’s how chemical bonds form. And if you get a bunch of atoms together in a solid, one electron can be shared among the whole solid. That’s the basis for the quantum theory of solids, which lets us understand how metals conduct electricity and how to make semiconductor computer chips. It’s all because electrons extend beyond specific planetary orbits.”
“Uncertain electrons are weird.”
“Strictly speaking, it’s not just electrons. Everything in the universe is subject to the uncertainty principle, and has an uncertain position and velocity.”
“That can’t be right. I mean, I can see my bone right over there, and it has a definite position, and a velocity of zero.”
“Ah, but the
quantum
uncertainty associated with your bone is dwarfed by the
practical
uncertainty involved in measuring it. If you look at it really carefully, you might be able to specify its position to within a millimeter or so—”
“I
always
look at my bone carefully.”
“—and with heroic effort, you might bring that down to a hundred nanometers. In that case, the velocity uncertainty of your hundred-gram bone would be only 10 -27 m/s. So, the velocity would be zero, plus or minus 10 -27 m/s.”
“That’s pretty slow.”
“Yeah, you could say that. At that speed, it would take the age of the universe to cross the thickness of a single atom.”
“Okay, that’s
really
slow.”
“We don’t see quantum uncertainty associated with everyday objects because they’re just too big. We only see uncertainty directly when we look at very small particles confined to very small spaces.”
“Like electrons near atoms!”
“Exactly.”
Uncertainty has another, even more profound effect on the structure of atoms. Electrons must always have uncertainty in both their position and momentum, and that means that the energy of an electron in an atom can never be zero. To have zero energy while still being part of an atom, an electron would need to be not moving, sitting right on top of the nucleus. This is impossible, as we’ve already seen—the closest we can come is to make a narrow electron wave packet centered on the nucleus, which will include lots of different states with nonzero momentum.Even the lowest-energy-allowed state of hydrogen, then, has some energy.
This is a general phenomenon, and applies to any confined quantum particle. If we know that a particle is in some particular region of space, that limits the uncertainty in the position, and increases the uncertainty in the momentum. Confined quantum particles are never at rest—they’re like puppies in a basket, always squirming and wiggling and shifting around, even when they’re asleep.
This tiny residual motion is called zero-point energy,
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