Understanding Quantum Mechanics: The Dual Nature of Particles and Waves

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In summary, the conversation discusses the fundamentals of quantum mechanics, specifically the concept of particles being viewed as waves and the interpretation of the wave equation. The wavefunction is the solution of the Schrödinger equation and contains all physical information of a quantum system. The observables, such as energy and momentum, are operators that work on the wavefunction, and squaring the resulting numbers gives the probability of the particle having a certain value. This concept is a fundamental difference from classical physics. The conversation also touches on the double slit experiment and the idea that a quantum particle is "nowhere" until a measurement is performed. The wavefunction is not the particle itself, but a description of its quantum state. The conversation ends with a reminder that
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StatusX
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In my last physics course, we spent the last couple of weeks introducing quantum mechanics, covering the schroedinger wave equation and simple examples like a particle in a box. So I don't know much about QM, and this may be a stupid question.

Anyway, we were taught that the particle can be thought of as a wave, described by the wave equation. Then they said the interpretation of the wave equation is that the integral of its magnitude squared over a region is the probablity of finding the particle in that region. So which is it? Is the particle some speck that can be anywhere, but is likely to be in the regions of high probability? Or is it the wave itself?
 
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  • #2
StatusX said:
In my last physics course, we spent the last couple of weeks introducing quantum mechanics, covering the schroedinger wave equation and simple examples like a particle in a box. So I don't know much about QM, and this may be a stupid question.

Anyway, we were taught that the particle can be thought of as a wave, described by the wave equation. Then they said the interpretation of the wave equation is that the integral of its magnitude squared over a region is the probablity of finding the particle in that region. So which is it? Is the particle some speck that can be anywhere, but is likely to be in the regions of high probability? Or is it the wave itself?

Hi, don't worry, your question is a very fundamental one and thus very important.

As you said, a particle can be viewed at as a wave. This vision is the main consequence of the double slit experiment where the "logic" adding of probabilities is no longer respected due to the interference term. I am sure you heard of this before.

deBroglie set up some relations (in his PhD-thesis) which connect the wavelike properties to the particle-like properties : E = hv and p = h/l where v is the frequency and l is the wavelength... E and p are energy and momentum of the particle...

Now the wavefunction is the solution of the Schrödinger-equation and this function contains all physical info of a certain QM-system. The observables like energy are now operators that work on this wavefunction. As a result of this you will get a number that is a possible energy-value. When you square this number you will get the probability that the system exhibits this energy-)value. This is a big difference with classical physics, now in QM the energy-value is not "really" important. It is the square that is important because it expresses the chance of the state having this particular energy-value. These energy-values are the socalled eigenvalues of the energy-operator. This operator is the Hamiltonian.

Just like this you need to see that the wave-function does contain all physical info, yet in order to acquire "numbers" describing the physical state you need to square the wavefunction. This gives you the probability of finding some particle at some given place...just like in the energy-eigenvalue-example, here above...

The reason for this fact (the square gives the probability of something) comes from the wave nature of this approach. In wave-physics, the square of the wave equation gives the intensity corresponding to the wave...This approach was also used in the double slit experiment and it turned out that the square of the wavefunction of the electrons gave the right probability-distribution that was observed at the detector-screen.
Well, keep in mind that this experiment was a thought experiment and cannot be executed in reality because the used apparatus would have to be extremely accurate...

regards
marlon

ps : i suggest you check this double-slit-experiment in your QM-textbooks or on the net. Just google away, there is enough available info out there...
 
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  • #3
StatusX said:
So which is it? Is the particle some speck that can be anywhere, but is likely to be in the regions of high probability? Or is it the wave itself?

Yes.

No.

Both.

None.

:eek:

Welcome to Quantum Mechanics !
:tongue2:

Hold your breath. A quantum particle is supposed to be described by a quantum state (your wavefunction, solution of the Schroedinger equation) and to be "nowhere" until a measurement of its position is performed.
Then the *probability* of finding the particle in a certain volume is given by what your professor told you (the square of the wavefunction, integrated over that volume). You're not supposed to ask where it is if you do not measure the position ; the punishment being incoherent answers.
So, yes, we've given up on saying where the particle is, or even on saying that the particle must be somewhere. One remark however: the particle IS NOT the wave function. The wave function is the state description of the quantum state of the particle. In the same way that the position and speed of a dust particle is not the particle itself in classical mechanics, but a description of its dynamical state.

cheers,
Patrick.
 

What is a probability wave?

A probability wave, also known as a wave function, is a mathematical representation of the likelihood of finding a particle in a specific location at a specific time. It is used in quantum mechanics to describe the behavior and properties of particles at the microscopic level.

How is a probability wave different from a regular wave?

A probability wave is a complex-valued function, whereas a regular wave is typically a real-valued function. Additionally, the amplitude of a probability wave represents the probability of finding a particle at a certain location, while the amplitude of a regular wave represents the intensity of a physical wave.

Why is a probability wave often represented by a complex-valued function?

The use of complex numbers allows for the inclusion of both amplitude and phase information in the wave function. This is necessary for accurately predicting the behavior of particles in quantum mechanics, as it allows for interference effects to be accounted for.

How does the measurement of a particle's position affect the probability wave?

When a particle's position is measured, the probability wave "collapses" to a single point, indicating the precise location of the particle at that moment. This is known as the collapse of the wave function and is a key aspect of the Copenhagen interpretation of quantum mechanics.

What is the relationship between the probability wave and the uncertainty principle?

The uncertainty principle states that it is impossible to know both the position and momentum of a particle with absolute certainty. The probability wave represents the uncertainty in a particle's position, and the mathematical description of the wave function is used to calculate the uncertainty in its momentum.

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