Schrödinger Potential Fields with no Energy Quantisation?

In summary: Continuum states, by the way, are essentially not a bound state anymore and in fact they don't correspond to physically realizable state because they do not go to zero at infinities (probably I should also have added this beforehand to the possible reason of the difficulty of observing continuum...).
  • #1
greswd
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The solution to the One-Dimensional Time-Independent Schrödinger equation for an electric potential field of constant value is an exponential function, and its energy eigenvalue can have any value, it is not quantised.

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Are there any other potential field functions whereby the energy of the particle is not quantised?

Excluding the case where the potential is entirely zero, i.e. free particles.
 
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  • #3
blue_leaf77 said:
The scattering case, i.e. a situation where the particle's energy is higher than the maximum value of the potential ##V(\mathbf{r})##. A particular example is the continuous solution of the hydrogen atom:
greswd said:
Are there any other potential field functions whereby the energy of the particle is not quantised?

very interesting. I didn't know that the atom could have a continuous spectrum. Why don't we observe this?
 
  • #4
We have observed this, starting many years ago:

"... in the exceptional case of the star AC +70 d 8247, surface gravity is about 3 million times the Earth's ... the hydrogen lines would be broadened to such an extent that they would flow together into a fairly uniform continuum of absorption. This might explain why there are white dwarfs with purely continuous spectra. According to Kuiper, AC +70 d 8247 has such continuous spectrum."

From "White Dwarfs", Otto Struve, Sky and Telescope, December 1953
 
  • #5
greswd said:
Why don't we observe this?
Probably some people out there have already observed this in collision experiments, but I don't know for sure. Anyway, continuum states are unbounded state. Given that an atom in reality is not floating alone in the universe, I imagine it must be hard to maintain a stable quasi-isolated hydrogen atom with positive energy without being quickly ionized.
secur said:
We have observed this, starting many years ago:

"... in the exceptional case of the star AC +70 d 8247, surface gravity is about 3 million times the Earth's ... the hydrogen lines would be broadened to such an extent that they would flow together into a fairly uniform continuum of absorption. This might explain why there are white dwarfs with purely continuous spectra. According to Kuiper, AC +70 d 8247 has such continuous spectrum."

From "White Dwarfs", Otto Struve, Sky and Telescope, December 1953
The continuous spectrum in the case of a star's emission sounds more like due to the usual line broadening mechanism, which is further due to a collective motion of an ensemble of atoms.
 
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  • #6
blue_leaf77, you're right. Struve says it's the "broadening of spectral lines when an element is influenced by an electric field" - due to other ionized atoms and electrons in the vicinity. Not the same as the continuous-spectrum case you're referring to, but seems close enough to greswd's request for "any other potential field functions whereby the energy of the particle is not quantized" to be worth mentioning.
 
  • #7
secur said:
but seems close enough to greswd's request for "any other potential field functions whereby the energy of the particle is not quantized"
Ah I see, so you intended to answer the original question of the thread. I thought you were responding to grewsd's last question in post #3 where the discussion is narrowing down to the issue of an isolated H atom's continuum states.
 
  • #8
blue_leaf77 said:
Given that an atom in reality is not floating alone in the universe, I imagine it must be hard to maintain a stable quasi-isolated hydrogen atom with positive energy without being quickly ionized.

What do you mean by ionized?
 
  • #9
greswd said:
What do you mean by ionized?
It's the ionization, the liberation of an electron from an atom following the addition of energy exceeding its binding energy.
 
  • #10
blue_leaf77 said:
It's the ionization, the liberation of an electron from an atom following the addition of energy exceeding its binding energy.

Is ionization an issue when examining the spectra of a hydrogen discharge lamp?
 
  • #11
greswd said:
Is ionization an issue when examining the spectra of a hydrogen discharge lamp?
You are drifting away from the current discussion. In post #5, I brought up ionization to present a possibility of the reason why the continuum state of hydrogen atom have been difficult to observe, I myself am not sure if there have been an observation out there that's why I also expressed my uncertainty in the same post. It has nothing to do with the spectrum of discharge lamp.

Continuum states, by the way, are essentially not a bound state anymore and in fact they don't correspond to physically realizable state because they do not go to zero at infinities (probably I should also have added this beforehand to the possible reason of the difficulty of observing continuum states).
 
  • #12
blue_leaf77 said:
You are drifting away from the current discussion. In post #5, I brought up ionization to present a possibility of the reason why the continuum state of hydrogen atom have been difficult to observe, I myself am not sure if there have been an observation out there that's why I also expressed my uncertainty in the same post. It has nothing to do with the spectrum of discharge lamp.

Continuum states, by the way, are essentially not a bound state anymore and in fact they don't correspond to physically realizable state because they do not go to zero at infinities (probably I should also have added this beforehand to the possible reason of the difficulty of observing continuum states).

Ok. So maybe there is possibly something added on top of the Schrödinger equation that excludes the physical possibility of continuum states?
 
  • #13
greswd said:
So maybe there is possibly something added on top of the Schrödinger equation that excludes the physical possibility of continuum states?
I don't think so, the reason why continuum states are not realizable is due to the non-vanishing wavefunction at infinities, and this is a consequence of satisfying the Schroedinger equation. Nevertheless, along with the discrete, bound states, continuum states can serve as the basis function of any realizable wavefunction.
 
  • #14
blue_leaf77 said:
I don't think so, the reason why continuum states are not realizable is due to the non-vanishing wavefunction at infinities

How does this non-vanishing prevent an electron in this state from converting its energy into a photon?
 
  • #15
When you say "in this state", do you mean one of the continuum state? Haven't I said that this state is not a realizable state.
 
  • #16
blue_leaf77 said:
When you say "in this state", do you mean one of the continuum state? Haven't I said that this state is not a realizable state.
Yes, I was asking why the non-vanishing wavefunction prevents it from being a realizable state.
 
  • #17
Because it's not normalizable. Pretty much like the wavefunction for a free particle.
 
  • #18
blue_leaf77 said:
Because it's not normalizable. Pretty much like the wavefunction for a free particle.
But a free particle can convert its energy into photons right?
 
  • #19
greswd said:
But a free particle can convert its energy into photons right?
In which way, if it's alone in the universe? Moreover about the "freeness" of that electron, its wavefunction in reality does not exactly equal to that of the theoretical free particle's wavefunction.
 
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  • #20
blue_leaf77 said:
In which way, if it's alone in the universe?

Good question. I don't even know how electrons in an atom lose energy, I just know that they do.
 
  • #21
greswd said:
I don't even know how electrons in an atom lose energy,
They can lose energy and drop down to a lower level by interaction with vacuum field (spontaneous emission), stimulated emission, and non-radiative energy transfer.
 
  • #22
blue_leaf77 said:
They can lose energy and drop down to a lower level by interaction with vacuum field (spontaneous emission)
So electrons in continuum states can't undergo spontaneous emission? Why not? Because they're non-normalizable?
 
  • #23
I'm a little confused about whether the original poster, greswd, considers his/her question answered? I think the answer is about the depth of the potential well. Depending on the shape of the well, we have the following possibilities for energy eigenstates:
  1. There are only continuum states. This is true if the well is too "shallow" to bind anything.
  2. There are energy bands of continuum states, and other energy ranges where there are discrete states.
  3. There are only discrete energy eigenstates.
The first situation holds for "shallow" potentials (or repulsive potentials). For example, [itex]V(x) = -A e^{-\lambda x}[/itex]. If [itex]A[/itex] or [itex]\lambda[/itex] is too small, I don't think that there will be any bound states.

The second situation holds for deep wells that don't become shallower with distance (for example, the Coulomb potential). [itex]V(x) = -A/r[/itex].

The third situation holds for deep wells that remain deep at large distances (for example, the harmonic oscillator potential). [itex]V(x) = -\frac{1}{2} k x^2[/itex]
 
  • #24
My first question has been answered. I do wonder why we don't observe the continuum states of hydrogen though.
 
  • #25
greswd said:
I do wonder why we don't observe the continuum states of hydrogen though.
How many times do I have to repeat, such a state is not normalizable.
 
  • #26
greswd said:
So electrons in continuum states can't undergo spontaneous emission? Why not? Because they're non-normalizable?

I wouldn't put it that way. I'm not sure what normalizability has to do with it. If you consider a free particle, with momentum [itex]\vec{p}[/itex], it's impossible for it to emit a photon for kinematic reasons: You have an initial momentum [itex]\vec{p}[/itex]. After emitting a photon, you have photon momentum [itex]\vec{K}[/itex] and particle momentum [itex]\vec{p'}[/itex]. For a free particle, there is just no way to choose [itex]\vec{K}[/itex] and [itex]\vec{p'}[/itex] so that both energy and momentum are conserved. If the particle is tightly bound (to a proton, for instance), then some excess momentum can be shared with the proton, and it becomes easier to satisfy the conservation laws.
 
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  • #27
blue_leaf77 said:
How many times do I have to repeat, such a state is not normalizable.
I asked the question only once before, in #14, and your answer in #17 is actually a repetition that doesn't answer my question.
 
  • #28
greswd said:
I asked the question only once before, in #14, and your answer in #17 is actually a repetition that doesn't answer my question.

What do you mean when you say that the continuum states are unobservable? When an electron is knocked out of its atom and becomes an unbound electron, that means that it has left the discrete states and entered into the continuum states.

[edit] That's not exactly right. The state of the electron after being knocked out of the atom is never a pure continuum state, since those are not normalizable. But its definite can be written as an infinite superposition of continuum states, in the same way that a function [itex]f(x)[/itex] can be written as a superposition of plane waves: [itex]f(x) = \int dk e^{ikx} \tilde{f}(k)[/itex].
 
  • #29
greswd said:
So electrons in continuum states can't undergo spontaneous emission?
Alright, when you said continuum state I always imagine you were talking about a particular continuum state. As I have pointed out, although such state is not normalizable, it can act as basis function to form a normalizable state. So, yes an electron can be found in continuum state around an atom if its in a certain linear combination of the atom's stationary states. But a free electron (here free means no other entity present in space except that electron) cannot suddenly lose energy for a reason which has been explained by steven above. However, in some circumstances an electron in continuum state around an atom or ion (so, this electron is not free but is not bound either) can join together into a bound state of that atom/ion by emitting photons of appropriate energy. An example of this phenomenon is the so-called high-harmonic generation.
 
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  • #30
stevendaryl said:
What do you mean when you say that the continuum states are unobservable? When an electron is knocked out of its atom and becomes an unbound electron, that means that it has left the discrete states and entered into the continuum states.
you already mentioned that its due to conservation laws
 
  • #31
greswd said:
you already mentioned that its due to conservation laws

That was the answer to the question: Why can't a free electron emit a photon? You can observe free electrons in other ways besides looking for photons that they emit.
 
  • #32
stevendaryl said:
That was the answer to the question: Why can't a free electron emit a photon? You can observe free electrons in other ways besides looking for photons that they emit.
Yeah, I was thinking from a photonic, spectroscopy POV
 
  • #33
Well, here's something to consider when thinking about spontaneous emission. If you have an electron in a high-energy state, then intuitively you understand why it would emit a photon and fall into a lower-energy state: It's natural for systems to want to lower their energy, in the same way that it's natural for water to want to flow downhill. But this intuitive answer doesn't actually make any sense, by itself. When an electron emits a photon, its energy goes down, but the energy in the electromagnetic field goes UP. The total energy is unchanged. So the real question is not: why does the electron's energy go down, but why does nature prefer to give its energy to photons, as opposed to electrons?

Well, we can understand that through entropy, which amounts to counting states. There is only one (or a small number) of ways that a bound electron can absorb a quantity of energy, because there are only a few states associated with a given energy. In contrast, there are infinitely many ways that photons can absorb a quantity of energy, because there are continuum-many photon states. So if you pick a way to split energy up between an electron and the electromagnetic field, it's overwhelmingly more likely that most of the energy will go to the electromagnetic field. So what we observe is that electrons tend to radiate their energy away.

Now, if the electron itself has continuum-many states, then this counting argument doesn't apply. Now, there is no good reason, as far as entropy, for the electron to give up its energy to the electromagnetic field.
 
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  • #34
Oh I see. A probabilistic way of looking at things.
 
  • #35
stevendaryl said:
For a free particle, there is just no way to choose [itex]\vec{K}[/itex] and [itex]\vec{p'}[/itex] so that both energy and momentum are conserved.

This is easiest to see in the rest frame of the particle. Then initially, the momentum is zero. Initially, the energy is [itex]mc^2[/itex]. After it emits a photon, the energy of the particle must still be at least [itex]mc^2[/itex] (because that's the lowest possible energy of the particle), which means that the energy of the photon has to zero (or negative!) to get energy to balance.
 
<h2>1. What is the Schrödinger Potential Field with no Energy Quantisation?</h2><p>The Schrödinger Potential Field with no Energy Quantisation is a theoretical concept in quantum mechanics that describes a system in which energy levels are not quantized. This means that the energy of the system can take on any value, rather than being restricted to discrete levels.</p><h2>2. How does the Schrödinger Potential Field with no Energy Quantisation differ from traditional quantum mechanics?</h2><p>In traditional quantum mechanics, energy levels are quantized, meaning they can only have certain discrete values. In the Schrödinger Potential Field with no Energy Quantisation, energy levels are continuous and can take on any value. This has significant implications for the behavior of particles and systems.</p><h2>3. What are the potential applications of the Schrödinger Potential Field with no Energy Quantisation?</h2><p>The Schrödinger Potential Field with no Energy Quantisation has potential applications in the study of complex systems, such as biological systems, where energy levels may not be well-defined. It may also have implications for the development of new technologies, such as quantum computing.</p><h2>4. How is the Schrödinger Potential Field with no Energy Quantisation related to the uncertainty principle?</h2><p>The Schrödinger Potential Field with no Energy Quantisation is related to the uncertainty principle in that it challenges the traditional idea that energy levels are well-defined and measurable. The uncertainty principle states that there is a fundamental limit to how precisely we can know certain pairs of physical properties, such as position and momentum, and this uncertainty is also applicable to energy levels.</p><h2>5. Is there any experimental evidence for the existence of the Schrödinger Potential Field with no Energy Quantisation?</h2><p>Currently, there is no experimental evidence for the existence of the Schrödinger Potential Field with no Energy Quantisation. It is a theoretical concept that is still being explored and studied by scientists. However, some experiments have shown behavior that is consistent with the idea of continuous energy levels, providing some support for this concept.</p>

1. What is the Schrödinger Potential Field with no Energy Quantisation?

The Schrödinger Potential Field with no Energy Quantisation is a theoretical concept in quantum mechanics that describes a system in which energy levels are not quantized. This means that the energy of the system can take on any value, rather than being restricted to discrete levels.

2. How does the Schrödinger Potential Field with no Energy Quantisation differ from traditional quantum mechanics?

In traditional quantum mechanics, energy levels are quantized, meaning they can only have certain discrete values. In the Schrödinger Potential Field with no Energy Quantisation, energy levels are continuous and can take on any value. This has significant implications for the behavior of particles and systems.

3. What are the potential applications of the Schrödinger Potential Field with no Energy Quantisation?

The Schrödinger Potential Field with no Energy Quantisation has potential applications in the study of complex systems, such as biological systems, where energy levels may not be well-defined. It may also have implications for the development of new technologies, such as quantum computing.

4. How is the Schrödinger Potential Field with no Energy Quantisation related to the uncertainty principle?

The Schrödinger Potential Field with no Energy Quantisation is related to the uncertainty principle in that it challenges the traditional idea that energy levels are well-defined and measurable. The uncertainty principle states that there is a fundamental limit to how precisely we can know certain pairs of physical properties, such as position and momentum, and this uncertainty is also applicable to energy levels.

5. Is there any experimental evidence for the existence of the Schrödinger Potential Field with no Energy Quantisation?

Currently, there is no experimental evidence for the existence of the Schrödinger Potential Field with no Energy Quantisation. It is a theoretical concept that is still being explored and studied by scientists. However, some experiments have shown behavior that is consistent with the idea of continuous energy levels, providing some support for this concept.

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