What makes a quantum state bound

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Wiktionary 4. Freebase 3. How to pronounce bound state? Alex US English. David US English. Mark US English. Daniel British. Libby British. Mia British. Karen Australian. Mathematically, this means that the solution must vanish at the walls:. We expect oscillating solutions, so the most general solution to this equation is. Applying the boundary condition expressed by Figure gives. Because we have , the solution must be.

If is zero, for all values of x and the normalization condition, Figure , cannot be satisfied. Assuming , Figure for then gives. We discard the solution because for this quantum number would be zero everywhere—an un-normalizable and therefore unphysical solution. Substituting Figure into Figure gives. A particle bound to a one-dimensional box can only have certain discrete quantized values of energy. To evaluate the allowed wave functions that correspond to these energies, we must find the normalization constant.

We impose the normalization condition Figure on the wave function. Hence, the wave functions that correspond to the energy values given in Figure are. For the lowest energy state or ground state energy , we have.

The index n is called the energy quantum number or principal quantum number. The state for is the first excited state, the state for is the second excited state, and so on.

The first three quantum states for of a particle in a box are shown in Figure. The wave functions in Figure are also called stationary state s and standing wave state s.

Energy quantization is a consequence of the boundary conditions. If the particle is not confined to a box but wanders freely, the allowed energies are continuous. However, in this case, only certain energies … are allowed. The energy difference between adjacent energy levels is given by.

Conservation of energy demands that if the energy of the system changes, the energy difference is carried in some other form of energy. For the special case of a charged particle confined to a small volume for example, in an atom , energy changes are often carried away by photons. A Simple Model of the Nucleus Suppose a proton is confined to a box of width a typical nuclear radius. What are the energies of the ground and the first excited states?

If the proton makes a transition from the first excited state to the ground state, what are the energy and the frequency of the emitted photon? Strategy If we assume that the proton confined in the nucleus can be modeled as a quantum particle in a box, all we need to do is to use Figure to find its energies and.

The mass of a proton is The emitted photon carries away the energy difference We can use the relation to find its frequency f. The first excited state:. The energy of the emitted photon is. Significance This is the typical frequency of a gamma ray emitted by a nucleus. The energy of this photon is about 10 million times greater than that of a visible light photon. The expectation value of the position for a particle in a box is given by. We can also find the expectation value of the momentum or average momentum of a large number of particles in a given state:.

Thus, for a particle in a state of definite energy, the average position is in the middle of the box and the average momentum of the particle is zero—as it would also be for a classical particle. Note that while the minimum energy of a classical particle can be zero the particle can be at rest in the middle of the box , the minimum energy of a quantum particle is nonzero and given by Figure.

The average particle energy in the nth quantum state—its expectation value of energy—is. The result is not surprising because the standing wave state is a state of definite energy.

Any energy measurement of this system must return a value equal to one of these allowed energies. By contrast the Voyager probes are barely unbound and will fly slowly off into the galaxy. Nevertheless "external potentials" are always used as a model of the interaction with something else, when you don't really care about the dynamics of the "something else". A very good example of this is how you derive the energy levels for an atom as the bound states of the electron in the external potential generated by the nucleus.

Pearson offers a physical interpretation for a related phenomenon, "singular continuous spectrum" Quantum mechanics is endlessly full of surprises.

Btw, I guess he forgot to say 'in three dimensions'. It would make this answer even more valuable. That's one ugly beast. Kosala Herath Kosala Herath 1 2 2 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Now live: A fully responsive profile. Linked 0. Related 4. Hot Network Questions. Question feed. Physics Stack Exchange works best with JavaScript enabled.

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