What is the Nature of Material Waves in Quantum Mechanics?

[I_am_learning]

So far I have learned these (correct me If I am wrong)
A photon can be considered both an wave and particle. The wave equivalent of the photon is the superposition of Electric and Magnetic Field Waves.
Similarly, De Broglie expanded this concept into matter. He proposed that Every matter has wave equivalent, and this wave is represented by a Wave function Psi.

But what does the material wave actually feel like?
Electromagnetic Waves can carry signal, What do material waves do??
Or are they just imaginary waves?

 

[I_am_learning]

What I am basically Asking is--What is the physical significance of Wave function of a particle?

 

[mikeph]

The essence is that a particle in quantum mechanics can be completely described by its wavefunction, so you can think of this function as a description of the state of that particle. The classical analogue is that, given the mass, velocity, charge, etc. numbers, you have all the information you need. In QM all you require is a single complex function whose domain is over space and time.

Although it contains all the information you might want, you have to do processing to obtain values that you want, for, say, the position, and these outputs are governed by the laws of quantum mechanics, so you will not get a result like "the positions is at (0,3,4), you'll get a probability density function which you can integrate over any particular volume to find the probability that the particle will be in that volume.

The wavefunction, or "state" of the particle is crucial to quantum mechanics, but without much experience of QM it's easy to see how someone wouldn't appreciate the value in this weird complicated view of a particle. The wavefunction is entirely mathematical, invisible, unobservable, but the results of the processing to it do match what we observe, and this explanation succeeds in predicting many phenomenon that classical physical fails to predict.

 

[jtbell]

What is the physical significance of Wave function of a particle?

If you mean, "what can we do with the wave function," we use it to calculate probabilities of the different allowed values for various physical quantities of the particle.

If you mean, "what the wave function really is," this is the subject of interpretations of quantum mechanics, about which there is no general agreement. In this forum you can find many threads debating the merits of various interpretations (de Broglie-Bohm pilot wave theory, many-worlds interpretation, Copenhagen interpretation, etc.).

 

[victorphy]

It is easier to understand if you regard the wave just as an imagination,it tells us where a particle will be when we try to find it,i.e in this case a particle just acts as a wave.But It does not really exist,I think.

 

[I_am_learning]

I have one more confusion.
Do de Broglie Waves mean that --> A moving particle really swings up-and-down (transverse to direction of motion) to exhibit a wave like behavior?

Or Is it that the de-Broglie Wave is just an imaginary and It is represented by a wave function (say Psi) whose square of amplitude is directly proportional to probability of finding the particle around that point?

If my second question's answer is Yes, Then how on Earth did de Borglie Came up with the idea that the probability of finding a moving particle around some points (points with amplitude 0 in the wave function) in its line of motion, can be totally 0?

 

[mikeph]

The second one. The particle doesn't actually oscillate in reality.

I am not sure about the relationship to wavefunction because at undergrad level we hardly did anything to do with the De Broglie wavelength, it seems just a relic of my earlier education. The wavefunction is the real thing (well... not actually real, but the fundamental thing), I just take the DB wavelength to be a representation of the 'scale' of the wavefunction, somewhat useful in working out how particles diffract. (I am probably under-selling it though)

I don't really know the origin of the QM theory, it does sound absurd, for example that a particle in a box has zero probability of being found exactly at the centre.

 

[Born2bwire]

I do not really recall deBroglie's motivation for his ideas, but it is my recollection that this was an immediate precursor to quantum mechanics. You can find deBroglie's Nobel Prize speech here:http://nobelprize.org/nobel_prizes/physics/laureates/1929/broglie-lecture.pdf I do not have the time to read it right now but maybe it will elucidate his reasoning. Either way, deBroglie waves were theorized before quantum mechanics was really fleshed out so do not take it to heart as how things really work. It's still useful because it is correct when we talk about a free particle.

Talking about particles being waves is a bit abstract because when most people visualize a wave they visualize a physical oscillation, like the disturbance of water in water waves or atoms and molecules in sound waves. Even classical electromagnetic waves are easy to grasp because we visualize the amplitudes of the fields to be oscillating. But the idea of a wave here is more abstract, the wave properties we are applying to the particle deal with the ideas like interference that we see in a wave. For example, water waves will add and cause constructive or destructive interference. In a quantum particle, the wave properties, contained in the wavefunction, will replicate interference. This interference is manifested in the probability density, the probability of finding a value (technically a range of values) when we make a measurement.

So, the obvious is the Young's double-slit. The interference of the wavefunction gives rise to a wavelike result when we plot the position of detected particles after passing through the slit. It is not that the particles are physically oscillating in space like a wave on a string (if we can be as bold to assume this), but that the interaction of the particles with themselves and each other follows wave behavior.

 

[I_am_learning]

a particle in a box has zero probability of being found exactly at the centre.

How is that?

 

[mikeph]

Because the wavefunction is like a sine wave, zero at either side of the box (cannot exist in the wall), so necessarily zero in the middle. The position operator multiplies the wavefunction by its conjugate so you get a sin^2 wave, which is zero still in the centre. So you have zero probability density there. You're most likely to find the particle at 1/4 or 3/4 across the box, because the confinement of the particle by the walls will set up the quantum analogue of a standing wave for the particle.

 

[jtbell]

Because the wavefunction is like a sine wave, zero at either side of the box (cannot exist in the wall), so necessarily zero in the middle.

For n = 1, 3, 5, ..., $\psi$ has a maximum magnitude at the center of the box. For n = 2, 4, 6, ... it's zero at the center of the box. See the graphs at

http://hyperphysics.phy-astr.gsu.edu/HBASE/quantum/Schr.html

 

[mikeph]

Ok, I referred to the lowest energy level without explicitly saying so.

 

[xepma]

The ground state also has a maximum right in the middle of the box.

Note that an oscillations does come into play if you consider a superposition of two energy eigenstates. In that case the wavefunction looks like:

$$\Psi(x,t) = c_1 e^{i E_1 t/h)\phi_1(x) + c_2 e^{i E_2 t/h)\phi_2(x)$$

and the absolute square (the probability density) contains an oscillating part:

$$|\Psi(x,t)|^2 = c_1^2 \phi_1(x)^2+ c_2^2 |\phi_2(x)^2 + 2c_1c_2 \cos((E_1-E_2)t/h) \phi_1(x)\phi_2(x)$$

(The eigenfunctions $$\phi_1$$ and $$\phi_2$$ are real valued, in this case, and for simplicity I've taken the coefficients $$c_1$$ and $$c_2$$ to be real valued as well).

As you can see, and this is a general result, as soon as a wavefunction is in a superposition of energy eigenstates, you always get an oscillations. The probability density will oscillate between the two eigenstates, with a frequency $$\omega = (E_1-E_2)/h$$

For a particle sitting in one energy eigenfunction there is, on a mathematical level, some oscillation going on: the overall (time dependent) phase of the wavefunction is $$e^{iEt/h}$$. This is an oscillation of the phase with frequency $$\omega = E/h$$. This phase cancels out when you take the absolute square.

 

[gabrielh]

Just throwing in a question of my own here. If I understand this correctly, the wavefunction of, say, an atom, is just a mathematical representation of the possible states of that atom. If the atom were to be moving in some direction and come into contact with a semitransparent mirror, part of the wave function is reflected and another part passes through. The reflected wave function ends up in box A and the wave function that passed through ends up in box B. With that said, (and correct me if my understanding is flawed), the wavefunction, being a probability of the possible location of the atom, in both boxes, which follows that since this wavefunction is in both boxes, the atom is in both boxes as well, until it is observed.

Does this show that the wavefunction is a physical thing? I'm just curious if this is an accurate discription of what happens.

 

[xepma]

In the end, all the wavefunction does is give you certain probabilities of where the particle can be found. So no, it's is not physical by itself. You cannot measure the wavefunction.

Note that through interaction with the environment an "observation" already takes place, so in most practical cases the particle really is in one particular box and the superposition is destroyed. This is called decoherence.

 

[gabrielh]

In the end, all the wavefunction does is give you certain probabilities of where the particle can be found. So no, it's is not physical by itself. You cannot measure the wavefunction.

Note that through interaction with the environment an "observation" already takes place, so in most practical cases the particle really is in one particular box and the superposition is destroyed. This is called decoherence.

Thank you. I'd like to follow with another question. If this environmental interaction counts as observation, under what conditions would the particle be in a superposition state? In other words, how must my scenario be altered to achieve a superposition state?

 

[jtbell]

Ok, I referred to the lowest energy level without explicitly saying so.

The lowest energy level has n = 1 and has a maximum at the center of the box. There's no n = 0 level.

 

[xepma]

Thank you. I'd like to follow with another question. If this environmental interaction counts as observation, under what conditions would the particle be in a superposition state? In other words, how must my scenario be altered to achieve a superposition state?

Some sort of control over the interaction of the particle with the environment. If you know how to do this, please tell me how ;)

 

[zonde]

If the atom were to be moving in some direction and come into contact with a semitransparent mirror, part of the wave function is reflected and another part passes through. The reflected wave function ends up in box A and the wave function that passed through ends up in box B. With that said, (and correct me if my understanding is flawed), the wavefunction, being a probability of the possible location of the atom, in both boxes, which follows that since this wavefunction is in both boxes, the atom is in both boxes as well, until it is observed.

I do not know about atoms and semitransparent mirrors but photon will be only in one box but still it will be in superposition because you can see it or miss it when you will observe it.
And in that sense wavefunction is physical thing.

 

[I_am_learning]

For n = 1, 3, 5, ..., $\psi$ has a maximum magnitude at the center of the box. For n = 2, 4, 6, ... it's zero at the center of the box.

Does a wave function being zero (therefore probability being 0) somewhere implies that---
---> the particle can't move past that point
--->or that the particle can't be in that point in any instant?
---> or that the particle is in that point for so small time that the probability of finding it there ** is infinitesimally small.

 

[jtbell]

It means that if you measure the position of the particle (or do something that effectively measures the position), you will not find it at that point.

The wave function $\Psi$ does not address questions about what the particle is "really doing, inside the wave function" before you measure it. This is again a matter of interpretation of QM, about which there is no general agreement at this time.

 

[I_am_learning]

It means that if you measure the position of the particle (or do something that effectively measures the position), you will not find it at that point.

Suppose in the particle in a Box scenario, The walls are some kind of sensors that records its counts, whenever the particle bounds off the wall.
Since the wave function is 0 at the walls, and you mean that it implies we can't find the particle there (The counter won't record any count),
Then Does this means that the particle bounds off the wall without touching the wall?

 

[jtbell]

You need to discard the picture of a classical-like particle bouncing between the walls of the box.

Also, you need to recognize that the "particle in a box" (more formally known as the "infinite square well") is an idealization, because no potential well has infinitely-"high" walls. For a silghtly more realistic situation, see the particle in a finite-walled box, in which the wave function is not zero at the walls.