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In summary, the conversation discusses the classical analog for the l=0 state, which is the state where the angular momentum is zero. It is noted that in classical mechanics, an orbit with zero angular momentum is not possible, but in quantum mechanics, it is observed as a superposition of unreflected and reflected paths. The conversation also touches on the effects of the centrosymmetric potential and the possibility of the electron speed exceeding the speed of light near the nucleus.
- #1
quantum123
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What is the classical analog for l=0 state?
Angular momentum = 0 , what kind of orbits is that?
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- #2
tejas777
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Classically you cannot have an orbit with zero angular momentum. Consider a non-quantized (or classical) version of the Bohr model. Check this for exact calculations:
http://en.wikipedia.org/wiki/Bohr_model
The centrifugal force experienced by the electron will be balanced by the electrostatic attraction it experiences from the nucleus. This force balance condition will give you the velocity of the electron as a function of the radius. This functional dependence goes like [itex]1/\sqrt{r}[/itex]. The angular momentum of this electron is given by [itex]L = mvr[/itex]. Therefore, the angular momentum depends on radius as [itex]\sqrt{r}[/itex]. Consequently, the angular momentum is zero when radius is zero. Hence we do not really have an orbit.
- #3
quantum123
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Thanks.
I find the l=0 state to be one that the quantum and classical pictures differ most strikingly.
- #4
DrDu
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For angular momentum zero, in a centrosymmetric potential, the particle will move from -r to +r and back again along a line of constant angle phi. Hence it falls through the center. I do not see why this should be classically forbidden. If an obstacle (like a nucleus) happens to be in the center, the particle may or may not get reflected. In classical mechanics, the particle either gets completely reflected or not reflected at all, while in QM (like in the hydrogen atom) you usually observe a superposition of unreflected and reflected paths. Furthermore in QM, the angle phi is undetermined, which does not mean that it changes in time.
- #5
tejas777
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DrDu said:
For angular momentum zero, in a centrosymmetric potential, the particle will move from -r to +r and back again along a line of constant angle phi. Hence it falls through the center. I do not see why this should be classically forbidden. If an obstacle (like a nucleus) happens to be in the center, the particle may or may not get reflected. In classical mechanics, the particle either gets completely reflected or not reflected at all, while in QM (like in the hydrogen atom) you usually observe a superposition of unreflected and reflected paths. Furthermore in QM, the angle phi is undetermined, which does not mean that it changes in time.
Yes, that is a good example.
- #6
quantum123
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Thanks.
Nuclear size being 10-15m, the electrostatic attraction tends to negative infinity. Will the electron speed exceed speed of light?
- #7
DrDu
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No, you have to consider relativistic corrections (i.e. the Dirac equation) in the immediate vicinity of the nucleus.
FAQ: L=0 State: Understanding Angular Momentum and Orbital Movement
What is the L=0 state?
The L=0 state refers to the quantum mechanical state of an atom or molecule where the angular momentum is equal to zero. This state is also known as the s-orbital or ground state, and it determines the shape and energy of the electron orbitals in the atom.
How is angular momentum related to orbital movement?
Angular momentum is a property of an object in motion that measures the amount of rotation it has around a specific axis. In the context of atoms, the orbital movement of electrons is related to their angular momentum, as the electrons are constantly moving in circular or elliptical paths around the nucleus.
What does the L=0 state tell us about the electron's energy?
The L=0 state, or the s-orbital, is the lowest energy state for an electron in an atom. This means that electrons in this state have the lowest possible energy level, and they can only move to higher energy states by absorbing energy.
How does the L=0 state contribute to the stability of atoms?
The L=0 state is crucial for the stability of atoms because it determines the shape and energy of the electron orbitals. These orbitals determine the arrangement and distribution of electrons around the nucleus, which ultimately affects the chemical properties and stability of the atom.
Can the L=0 state change?
Yes, the L=0 state can change if the atom absorbs or emits energy. This can cause the electron to move to a higher or lower energy state, which will change the shape and energy of the electron orbitals. However, the L=0 state will always be the lowest energy state for an electron in an atom under normal circumstances.
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