# Help with understanding dual nature of electron.

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## Main Question or Discussion Point

Hello,
I had read about the dual nature of electrons and the quantum numbers for some time ago and was always confused.
1-What does dual nature of electron mean? Does it mean that the electron moves in a sort of wave like motion around the nucleus? Where do the sub-shells and orbitals fit in?
2-Is there any diagram or simple explanation that shows where in the shells the sub-shells (s,p,d,f) exist and then the orbitals?

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tiny-tim
Homework Helper
Hello PrincePhoenix!
1-What does dual nature of electron mean? Does it mean that the electron moves in a sort of wave like motion around the nucleus?
Almost … it means that the electron is a sort of wave around the nucleus.
Where do the sub-shells and orbitals fit in?
They're like the individual frequencies or resonances in, say, the wave for the noise from a church bell.
2-Is there any diagram or simple explanation that shows where in the shells the sub-shells (s,p,d,f) exist and then the orbitals?
A quick google image-search shows http://www.chemistry.nmsu.edu/studntres/chem111/resources/notes/atomic_orbitals.html" [Broken] …

if you search for longer you may find some better ones.

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Gold Member
Almost … it means that the electron is a sort of wave around the nucleus. :wink
You mean something like this. (I'm not good at graphics but this is roughly what I mean)
[PLAIN]http://princephoenix.webs.com/untitled.bmp [Broken]

A quick google image-search shows http://www.chemistry.nmsu.edu/studntres/chem111/resources/notes/atomic_orbitals.html" [Broken] …

if you search for longer you may find some better ones.
I've seen those in my textbooks. I mean a diagram that shows them in perspective with the nucleus and the general structure of the atom.

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tiny-tim
Homework Helper
You mean something like this.
No, that's one-dimensional, I was thinking of a 3D wave.

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Okay a three-dimensional, but is it just its path of motion? I can't visualize it as a particle and a wave any other way.

tiny-tim
Homework Helper
Okay a three-dimensional, but is it just its path of motion? I can't visualize it as a particle and a wave any other way.
Particles have paths. Waves just are.
No idea.

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Thank you

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Is this accurate?

tiny-tim
Homework Helper

are those the 95% probability limits?​

It might help to think of the Quantum Nature of particles according to the following points:

• The particles are NOT waves. They are truly point particles. During ionization of atoms, no one has ever obsereved, for example, half an electron to emerge. It is always an integer multiple of electrons;
• You cannot know both the position and the velocity of the particle at a given instant in time. As such, the concept of trajectory is meaningless in Quantum Mechanics. There is only a probability law that tells us the probabilities of finding the particle at a particular point at later times provided that it was known to be at some position at a previous fixed instant in time;
• This probability is given by the square of the absolute value of a complex quantity called transition amplitude. It is this quantity that plays the fundamental role in the dynamics of quantum objects, like electrons, for example. Namely, according to the laws of conditional probability, if we know the conditional probability that the particle will go from A to some intermediate point C and then from C to B, then the conditional probability for the particle to go from A to B is given by the sum:

$$P(A \rightarrow B) = \sum_{C}{P(A \rightarrow C) P(C \rightarrow B)}$$

where the sum goes over all possible intermediate positions. However, this rule does NOT hold for probabilities, but for probability amplitudes in Quantum Mechanics:

$$T(A \rightarrow B) = \sum_{C}{T(A \rightarrow C) T(C \rightarrow B)}$$

and the probability is given by:

$$P(A \rightarrow B) = |T(A \rightarrow B)|^{2}$$
• It is easy to see that there is nothing special about one intermediate state in the above sum and the same law holds if we insert an arbitrary number of intermediate states. Specifically, if this number tends to an uncountable infinity, we go from an infinite number of infinite sums to a so called path integral. The transition amplitude from point with position vector $\mathbf{x}'$ at time $t'$ to a point with position vector $\mathbf{x}$ at time $t$ is denoted by $K(\mathbf{x}, t; \mathbf{x}', t')$ and it may be shown that is given by:

$$K(\mathbf{x}, t; \mathbf{x}', t') = \int{D\mathbf{x}(t) \, \exp\left[\frac{i}{\hbar} \, S(t, t')[\mathbf{x}(t)] \right]}$$

where:

$$S(t, t')[\mathbf{x}(t)] = \int_{t'}^{t}{L(\tilde{t}, x(\tilde{t}), \dot{\mathbf{x}}(\tilde{t})) \, d\tilde{t}}$$

is the action of the particle with a Lagrangian $L(t, \mathbf{x}, \dot{\mathbf{x}}$;
• If the Lagrangian is given by:

$$L = \frac{m \dot{\mathbf{x}}^{2}}{2} - V(\mathbf{x}, t)$$

then one can proove that the transition amplitude satisfies the following PDE:

$$i \hbar \frac{\partial K(\mathbf{x}, t; \mathbf{x}', t')}{\partial t} = -\frac{\hbar^{2}}{2 m} \nabla^{2}_{\mathbf{x}} K(\mathbf{x}, t; \mathbf{x}', t') + V(\mathbf{x}, t) K(\mathbf{x}, t; \mathbf{x}', t'), \; t > t'$$

with the initial condition:

$$K(\mathbf{x}, t; \mathbf{x}', t) = \delta(\mathbf{x} - \mathbf{x}')$$

The last partial differential equation is known as the time-dependent Schrodinger equation, but now it holds for the propagator of the particle. Since in classical mechanics only phenomena associated with the propagation of waves obey such types of partial differential equations, one might be tempted to interpret the propagation of the wave associated with the probability amplitude of the particle as the property of the particle itself. This, however, is not strictly true.

Feynman on Wave Particle Duality

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One often talks about the wave-particle duality of the electron meaning: in some experiments the electron behaves like a wave and in others like a particle.

1) http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/davger2.html#c1"
You shoot electrons at a crystal and observe an interference pattern. The pattern can be explained if you assume that the electron has a wavelength.

2) http://www.hitachi.com/rd/research/em/doubleslit.html" [Broken])
You observe dots on the screen (particles) but the pattern looks as if produced by wave interference.

http://www.perimeterinstitute.ca/Perimeter_Explorations/Quantum_Reality/Chapter_2_-_Wave-Particle_Duality_with_Electrons/" [Broken] (Perimeter Institute for Theoretical Physics)
The video can be viewed http://www.perimeterinstitute.ca/en/Perimeter_Explorations/Quantum_Reality/Suggested_Ways_to_Use_this_Package/" [Broken] I recommend the Quicktime video. The first half of the video deals with the electron double slit, the second half with photon double slit, molecule interference, different interpretations (pilot wave theory, Many Worlds, Copenhagen) and applications of quantum mechanics (for example quantum cryptography).

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Thanks to everyone for their help.