# How does a rectangular potential barrier work (in quantum mechanics)?

I'm having a hard time grasping the concept of 'potential barrier'. I'm currently concerned about a rectangular potential barrier (I have attached a picture).

The classical analogy given is of a ball rolling up a hill (I have attached a picture). In that case the ball starts losing energy as soon as it starts climbing the hill, and then goes on moving with the remaining energy. Here, the ball continuously loses energy for some distance $$d$$ which is the width of the hill? So, we can interpret it as a 1D motion ($$x$$-component of motion), where the potential barrier is consuming the energy continuously.

But in case of an electron going through the potential barrier (as shown in picture), which is correct?

• Does the electron lose some energy (equal to potential barrier) as soon as it enters the barrier, and then goes on with remaining energy through the remaining barrier?

• Does it continuously lose energy as it is moving through the barrier, until it crosses the barrier?

If the potential barrier has potential 'V' and the electron has energy 'E', then will this 'V' energy be lost by the electron in continuous increments until it passes the barrier and has energy E-V at the end?

• Will it instantly lose 'V' energy as soon as it enters the barrier?

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• "Here, the ball continuously loses energy for some distance d which is the width of the hill?" - I guess part of the confusion might be coming from you equating the wrong elements in these two images. For the classical image, the width $d$ does not correspond to the length L. The hill should look more like going up, then going back down to the same level - like a plateau. The rectangular barrier in the QM case is the same thing, but the hill's slope is idealized to be completely vertical. See slides 4 and 5 here Commented Apr 21, 2023 at 1:56
• In other words, $d$ is not the width of the hill, it's just the width of the transition region. Commented Apr 21, 2023 at 1:58
• Thanks for correction. So, in that sense, the electron looses all the energy(equal to potential of barrier) as soon as it enters the barrier region? Since we have taken an idealised vertical potential. Commented Apr 22, 2023 at 6:52
• I have added a correct version with the 'd' described by you. Commented Apr 22, 2023 at 7:01

The view you wish to get is impossible. All those pictures you are drawing are energy eigenstates of the Hamiltonian operator. That is, the entire wave, including the parts inside the barrier and through the barrier to the other side, must be together, before the electron has one single energy value. You therefore cannot think of it as part of the wave as having so and so energy. It is the whole thing that has one value of energy.

The correct classical situation to try and make sense of this, is light propagation. Where the potential is zero (i.e. left and right extremes), the light is moving in glass blocks. The potential barrier is an air gap. This light is supposed to undergo total internal reflection because the angle of incidence is greater than the critical angle (ignore the fact that this is a 1D calculation, this is the only sensible classical situation we can see this with our own eyes, so we have to somewhat stretch it to fit).

But because of "evanescent waves", the exponential decay of the light wave in the air gap can be interrupted by another glass block, and then the light could penetrate the classically impossible places, and tunnel over to the other side.

You may not speak of losing energy. The colour of the light stays the same, so it has the same photon energy. There is merely a relationship between the amplitudes of the oscillations.

• That's the correct answer, but I think the OP still has a valid case of confusion. Why are we plotting psi in this particular case? Shouldn't we be plotting the magnitude of psi squared? That would be a much better way of showing that the probability of finding the quantum is constant on each side of the barrier (Or is it? Reflections?) because in those regions we are talking about plane wave solutions. I think you mentioned that you are an educator? Do you remember why this is such an anomaly in the treatment of QM solutions? What am I missing about the way the problem is being plotted? Commented Apr 20, 2023 at 14:25
• It is meaningful to plot psi; it is equally meaningful to plot the electric field strength in the wave propagation of light. If you want to plot the magnitude of psi squared, that is possible but definitely much less insightful because the waviness of the solution will go away, and you will be unable to see the wavelength (which contributes to the energy term). The probability of finding the quantum is not the same on the two sides. Commented Apr 20, 2023 at 14:35
• Psi is complex, though, and the oscillating solution to the Schroedinger equation is not the physical solution. The physical solution is a plane wave coming in and being reflected on the barrier, which means that there should be an oscillating component on top of a constant for |psi|^2. Alternatively one could calculate the scattering coefficients and take this as the most simple entry level toy problem to what we are doing in quantum field theory. Now that would make sense. To show students this plot does, IMHO, not serve much of a purpose in terms of physical intuition building. Commented Apr 20, 2023 at 22:06

Quantum tunneling is a purely quantum effect and trying to phrase it in classical physics terms will give you unsatisfactory/uncomplete results.

One handwavy way to think of this is to remember that the wavefunction is always in superposition of many momentum states (unless you have a plane wave, but you will never meet those in nature). Some of the momentum states have enough energy to pass the barrier and so they will.

But there are some issues with this picture. We can look at the transmission $$T$$, which is basically the probability to tunnel trough the barrier. The probability of tunneling goes down as the barrier gets wider, which can't be explained with this picture. Also, for some values of $$E$$ (the energy of the state) the transmission goes down, which also can't be explained. In this picture below, $$a$$ is the potential width. So the red line is the widest barrier.

Image 1: By Becarlson - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=67889226

Image 2: By Д.Ильин: vectorization - File:TvsE9.tif by Drchicken1, CC0, https://commons.wikimedia.org/w/index.php?curid=106025611

• The Gaussian wave packet is a horrible way to illustrate quantum mechanics. 99% of your audience will mistake the wave packet for a quantum, which is absolute nonsense. That we don't do plane waves is also not true. Basically all of quantum field theory is based on the idea of scattering of plane waves on each other. Plane waves do have problems (normalization becomes non-trivial, you get IR divergence etc.), but they are actually the natural way of looking at QM. We are, after all, also not measuring |psi|^2 of atoms in the lab, We are measuring the scattering of light with a spectrometer. Commented Apr 20, 2023 at 22:11
• Tunneling is quite well understood in QM. If the energy of the impacting particle is less than the barrier, then the wave function is a decreasing exponential inside the barrier. The decay constant is proportional to the height of the barrier minus the particle energy. The decay is exponential in the width of the barrier. With a little more care, and a lot more math, you can calculate such things as the rate a molecule spontaneously decays from a local minimum energy. Commented Apr 21, 2023 at 18:24

# Classical vs quantum mechanical electron

You are thinking about the electron as if it was sometimes a classical particle, sometimes a classical wave. An electron is neither, though there are similarities. Thinking about the similarities makes life confusing in this case. Instead, you need to focus on what an electron actually is. It is like nothing you have encountered in everyday life.

In classical physics, an electron is a small point-like particle. It follows a trajectory. A force acts smoothly to change the trajectory. You could measure position and momentum at any time you like to arbitrarily good precision without disturbing the trajectory.

By contrast, in quantum mechanics, the effect of the outside world on an electron is often better described by discrete interactions. We may know a measured value before hand. We can measure it again afterward. But we don't see what happens during an interaction. These kinds of interaction change the state of the electron, but they can tell us information about the the electron. We can use them to make measurements of the electron.

A typical such interaction would be to shine a light on an electron and learn things by how it is reflected. But light comes in lumps called photons. Bouncing a photon off an electron can tell you about the electron, but it also changes the electron's energy and momentum.

You can use a short wavelength photon, which can be well localized. This will tell you more precisely where the reflection occurred. But a short wavelength photon is high energy. It gives the electron a strong kick that can't be very precisely determined. So you don't know the electron's momentum or energy very well afterward.

You can use a long wavelength to make the kick as gentle as you like. But you don't know very accurately where this photon is. You can't learn as much about where the electron is.

This inverse relationship between accurately knowing position and accurately knowing momentum turns out to be fundamental, not just a limitation in measurement. It is one of the reasons why waves are used to describe reality. An electron does not have a precise position or momentum. It always has a range of possible positions and momenta.

These ranges are different from anything classical. A bag of gas doesn't have a definite size or shape, and is always spread out to some degree. Some parts of it can go fast and others slow. But everywhere inside the bag has some definite amount of gas and that part has some definite momentum. An electron is not like this.

You can shoot an electron through vacuum to a phosphor covered glass screen. If you prepare the electron in a spread out state, the electron has some presence everywhere in the vacuum chamber. You know this because it can hit anywhere on the screen with equal probability. But it is wrong to think that everyplace in the chamber has some piece of electron. When the electron hits the screen, it hits one phosphorus atom and make it give off light. The other atoms are not disturbed. If you repeat the experiment, you will find spots of light uniformly distributed over the screen.

It is also wrong to think that the electron just is a particle, and you learn where it is when it hit the screen. In the spread out state, it does not have a position. There is no way to predict which atom will be hit. If you put two slits in the way, the electron would go through both slits at once and would interfere with itself like a wave on the other side. You would still find one electron lights up one atom. But the distribution would would be concentrated where interference added and less where it cancelled.

An electron always has a range of possible positions. That range can be as big as a vacuum chamber or as small as an atom. It can be far smaller. No experiment has found a limit as to how small. But it cannot be 0. If the position range is small, the momentum range is necessarily big. The electron in this state does not have a speed. There is no way to predict how long it will take to travel somewhere. It has a range of speeds and you can predict a range of times.

# An electron has no parts

An electron is something like a point particle. If it has a size, the size is smaller than any experiment so far can measure.

An electron has no parts. If an energetic charged particle comes flying by an atom, it might knock the electron out of the atom, or kick it into a higher energy orbital, or it might miss entirely. It never kicks half an electron out or kicks half an electron into another orbital. No matter how small or well-localized it is, the particle always interacts with an entire electron or no electron.

Also, we never see one part of an electron repel another part of itself. The motion of a free electron is guided by the Schrodinger equation. That is, it is attracted to regions of low energy, and tends to continue moving in the direction it is going as required by momentum. There is no term in the Hamiltonian for Coulomb repulsion with itself.

So this range of positions is places a point like electron might be, in a way totally different from anything classical. A classical point is always in exactly one position. A quantum electron is not.

In the double slit experiment, the wave function describes the degree of "presence" an electron has at each point in the vacuum chamber. You must take into account all of these positions and their phase to predict where the electron will be next.

The wave function says the electron has some "presence" in both slits. This does not mean a point like particle follows a trajectory through both slits at the same time. An electron has similarities to a point like particle, but it is something different than anything you see in everyday life. It has no measurable size and no parts. It has no position and no trajectory. It has a range of positions.

It is tempting to think of it as something like an ocean wave. This is misleading. An ocean wave has many parts, each in a definite location. A quantum wave describes where an electron has a "presence".

The idea of "presence" is illustrated when the electron hits the screen. The wave function has crests and troughs. The electron does not turn out to be everywhere where a crest is. It interacts with one atoms. Unlike classical mechanics, you cannot predict which atom will be hit. There are only probabilities. The probability is high where the wave function is large.

This is very different from classical ideas of cause and effect. Using classical thinking to understand it leads to more confusion.

# The potential barrier

Apply this to the potential barrier. As the the electron, approaches the barrier, it has some "presence" on the left side.

As it encounters the barrier, the wave is partially reflected and partially transmitted. The transmitted part says how much "presence" the electron has in the barrier, and later to the right of the barrier.

A high barrier is more strongly reflecting. The electron has a smaller "presence" in the barrier. This half-way makes sense if you think classically. It is classically impossible for the electron to be in the barrier. It is even more impossible if the barrier is higher. This kind of thinking feels like it leads you in the right direction. But it is always accompanied by confusion.

One way of handling the confusion is the famous quote "Shut up and calculate." This means that however confusing and contradictory to classical mechanics the concepts are, the equations correctly describe the behavior of the electron. The equations are more important. Working with them not only leads to solved problems, but also conceptual breakthroughs and new physics. Working with the concepts is much less productive.

But you can also get used to the concepts. In time they do seem more natural. For this, it helps to think as correctly about the concepts as possible, and to recognize where they are different from classical concepts.

# A more physical view

A more physical view of the barrier may help. The square potential is a good problem because the math is easy enough to solve, and it illustrated the behavior well. But it might be helpful to think about another barrier.

In classical physics, a point electron can approach an infinite sheet of charge. The sheet repels the electron. If the electron doesn't have enough kinetic energy, it stops and flies away.

In classical physics, this is described by forces between electrons in the sheet and our electron. Life gets simpler if the force calculation is divided into parts. The sheet generates an electric field, and the field generates the force. The field is really just a shortcut. It is a region of space where a charge would feel a force if it was there.

You can take this shortcut further with potential energy. An electron would have this much potential energy if it was at this point.

Quantum mechanics almost forces this shortcut on you. To calculate forces directly, you need to know how far our electron is from each electron in the sheet. (If the sheet is uniform, the field is uniform and you don't need to know.) But an electron has a range of positions. Fields work much better with wave functions.

Also this removes other particles from the problem. Elementary quantum mechanics is always about single particles and how they respond to force fields. And these force fields are always expressed in terms of potential energy.

As a classical electron approaches a sheet of charge, it gains potential energy and loses kinetic energy. As it passes through the sheet, you could add up the forces to find its energy. And similar on the other side. The electron follows a trajectory and is always at a single point at a given time.

In quantum mechanics, a wave function describes where the electron has a degree of presence, not where it is. To calculate the expectation value of the energy, you need to integrate over the wave function.

As the electron approaches the barrier, it more and more is contained in a region where the potential energy is high. The expectation of kinetic energy becomes low. It is possible for the electron to have a partial presence inside the barrier or on the other side.

If the electron interacts with an atom, the range of positions becomes small. There is no problem if the electron is outside the barrier.

If the electron interacted with an atom inside the barrier, it would be entirely inside the barrier. Given this much, you conclude that this is impossible. Energy is conserved. The electron can't interact with atoms inside the barrier.

But atoms inside the barrier have energy. For example, they have thermal motion, or they can have excited electrons. If it is possible for this energy to go into the interaction, the electron can be found at an atom inside the barrier.

• Hooooly! Thanks a loooot for such a long and elaborate answer. I've never read something this keenly before. I didn't understand the whole of it, but I got the gist of it. I will come back again sometime to read your answer again :) Commented Apr 20, 2023 at 17:38
• One thing you must tell me though. I have read a Feynman quote " the wave function for an electron in an atom does not describe a smeared out electron with smooth charge density. The electron is either here or there or somewhere, but wherever it is, its a point charge " This always makes me think that the wave function is telling me probability of finding this point particle i.e it is a particle in the end, its just that its 'presence' is known but its exact location is not, but there is a particle on some exact location. Commented Apr 20, 2023 at 17:45
• An electron is not a thing, to begin with. It is a quantum of energy, momentum, angular momentum and charges. We teach this properly in high school when we are discussing the photoelectric effect with photons as quanta of light. Unfortunately all of that goes out the window when your QM 101 professor starts talking about "particles". Particles don't exist. The only challenge for the professional physicist is that he has to understand that quanta are irreversible energy exchanges. There are no quanta in the free fields. That may be the only thing that is borderline hard to explain. Commented Apr 20, 2023 at 22:14
• @RohitShekhawat I used to tell people to read Feynman about QM. I don't do that anymore. Feynman is simply wrong. Quanta like an electron don't have a position property to begin with and they are not point charges. Quanta are amounts of energy, momentum, angular momentum and charges. Because these are conserved quantities people like to associate them with "objects". It's like we are associating money with coins and bank notes. Your banker doesn't. To him "money" is an accounting quantity that never gets lost. THAT is what an electron really is. Commented Apr 20, 2023 at 22:18
• @FlatterMann Wow! This money analogy is really good, its hits hard. So, the electron is not a 'particle'? But when we talk about probability distribution, who's probability are we talking about? When we say 'probability of finding a particle'(which is a wrong statement based on what you said), what are we really talking about? Commented Apr 22, 2023 at 6:38

Quantum tunneling is a purely quantum effect and it is described by the solution of the time dependent Schrodinger equation.Suppose we have a particle which evolves according to the time dependent Schrodinger equation in a potential:

$$V(x)=0$$ when $$-L , $$V(x)=c$$ when $$x=-L,L$$ and initial conditions $$\Psi(-L,t)=\Psi(L,t)=0$$ and $$\Psi(x,0) =g(x)$$ The solution of the Schrodinger equation for $$\Psi(x)$$ will tell you that at some finite time $$t$$ for $$|d|>|L|\rightarrow$$ $$|\Psi(d,t)|^{2}=h(t)>0$$ so the electron has jumped the barrier which in classical mechanics is impossible.

So it all comes down to the solution of Schrodinger's equation for a particle in a box as we call it,it is predicted by the solution of a partial differential equation.

• The electron is not an object that jumps. What quantum mechanics tells us is the probability of finding the incoming energy on the other side of the barrier. Nothing here "moves". Instead energy flows. One could easily represent this properly in classical mechanics with an energy density. It would be positive on the left hand side of the potential barrier and zero on the right hand side. We don't do that, which in terms of learning makes it much harder for the student to make the move from the classical to the quantum picture. One should also teach scattering coefficients on this example. Commented Apr 20, 2023 at 22:26
• @FlatterMann I dont think anything you have posted is correct LOL. Commented Apr 20, 2023 at 23:36
• You are welcome to show me an electron object. Despite the many trillions of measurements that high energy physics detectors that were partly designed by me have made, not once did those experiments detect a particle. What they were and still are detecting are all irreversible energy transfers. What is strange is that we teach this correctly in high school where we label photons "quanta of energy". It's in undergrad physics that the misleading "particle" language takes over the imagination and for most people it stays with them for their entire lives. Commented Apr 20, 2023 at 23:52
• This answer would benefit from a little more explanation of quantum tunneling. For example, for a barrier higher than the energy of the particle, inside the barrier the wave function decreases exponentially. Thus it has a finite chance of getting through, but that chance drops very rapidly as the barrier gets thicker. Thus quickly approaching the classical limit. Commented Apr 21, 2023 at 18:14