As we all know, the wavefunction of such a particle has a certain number n of zeros due to boundary conditions. If at these points the wavefunction is zero, then, since the probability of finding the particle there is equal to the square of the wavefunction, it follows that the particle cannot ever be there. However, there is nothing physical at those points that would prevent the particle from being there at some instant.
Moreover, a wavefunction psi_n corresponds to an energy level E_n. As you change to a higher energy level, the index n grows, and we have more nodes of the wavefunction; i.e., more places where the particle cannot be. Again, there is nothing physical at these points.
Note that you only focus your mind on what is called "elementary solutions" of a PDE (Schödinger's equation), however, the "physical" solution is usually (for linear problems) a superposition of all elementary solutions, leading to something realistic. The same situation you encounter in classical Physics, where elementary solutions being complex-valued or containing singularities are used, but nobody ever asks if this is physical or not - because everyone has got used to this!
In your case, you have to consider so-called "coherent states", superpositions of eigenstates of the hamiltonian operator which are very close to classical particle distribution.
In other words: your have to consider the forrest as a whole, and not each single tree of its own!
By the way: diffraction patterns of electron beams through a pinhole is a good experimental example for a "real-world" phenomenon with nodes!!
I don't understand what's mysterious about this . . . Suppose you have a classical wave, like a literal wave in a linear trough full of water; if you shake the trough at the frequency of n=2, there will be a standing wave with zero amplitude at a point halfway through the box. There is nothing physically in the trough that prevents the surface of the water from moving up and down at this point. It just comes out of the physics of the wave itself.
So if we can get past the part about why p
The node is point where the wave function is crossing the x-axis. If you give zero index to represent the ground state wave function, which doesn' cross the x-axis, the number of nodes tells us that we have a wave function for the nth existed state. Similarly Wo and Wn correspond to ground and n existed state energies.
Note that you only focus your mind on what is called "elementary solutions" of a PDE (Schödinger's equation), however, the "physical" solution is usually (for linear problems) a superposition of all elementary solutions, leading to something realistic. The same situation you encounter in classical Physics, where elementary solutions being complex-valued or containing singularities are used, but nobody ever asks if this is physical or not - because everyone has got used to this!
In your case, you have to consider so-called "coherent states", superpositions of eigenstates of the hamiltonian operator which are very close to classical particle distribution.
In other words: your have to consider the forrest as a whole, and not each single tree of its own!
By the way: diffraction patterns of electron beams through a pinhole is a good experimental example for a "real-world" phenomenon with nodes!!
Two successive nodes determine the region where the probability to find a particle in the midway is very high. For a particle in a box, The number of nodes plus one, is called the principal quantum number since it fully describes (labels) each state in this one dimensional example.
Good explanation, Markus; however, I still have to think it over. Thank you.
I'm sorry, Henk. I didn't mean to be pedantic. Quite the contrary: since I am quite ignorant of Physics, I am always afraid of alluding to a well-known property as if it were a great novelty, hence my usage of "as we all know...".
That is similar to the Young's slits. In the interference pattern there are black line, although there is nothing physical at that place of the screen that prevents the particle from being detected there. That has been verified experimentally.
All these physical arguments rely on our mental picture how we persive the physical world surrounding us plus the classical teaching of physics. Clearly miss leading otherwise our grand fathers wouldn't propose a new physics, which has many mathematical representations (wave mech., quantum mech. etc.) plus the Neil Born statistical perception of reality if of course exist any.
I gave the clear and concise mathematical meaning of nodes , which arise in the solutions of very special type of DE involving characteristic parameter, '' Existence of the solutions is possible if only if that parameter takes a set of distinct numbers, even continues ' plus its probabilistic interpretation by Neil Born, which Einstein rejects categorically by saying that GOD doesn' play gambling, but uses it if he has too!!!
Maybe the key point you thinking about "can't be founded" has some defect
Classical physics talks force where you adopt to explain the possibility in quantum physics , which i think,is an initial defect.
Second,some model like 1 dimensional infinite potential well is academic,not realistic,thus the singularities may not actually exist.
Last,can't be founded by measure does't mean
Markus says superpositions of eigenstates, I highly agree so.
Forgive my English is not good.
A minor point: it is easy to get confused by the expression ``probability to be *at* a given point''. I prefer to think in terms of the probability of being close to a given point x, say, within the interval [x-l/2,x+l/2] with l small. Then the difference between the nodes and ordinary points is the following: for ordinary points, the probability to be in such an interval decreases to zero as l, whereas for a node it decreases as l^3 (because the probability is |psi(x)|^2). So you are right in saying that the nodes are points, where the probability of finding the particle is low, but it is misleading to say that tthe probability is zero, in particular because the probability of being in any specific point is zero as well.
Dear Tarik, I meant no criticism. But indeed your very text shows the difficulty I was alluding to: you say ``Now assume that I am an ordinary man just you met on the street , and I ask you what is the probability to find me just here? x=xo, '' This probability is zero. Proof: let the street run from zero to one. I can divide this length in n segments of length 1/n. Now xo belongs to one and at most one of these segments, say the one from k/n to (k+1)/n. But the probability to belong to one of n segments is 1/n if the ``ordinary man'' is with equal probability anywhere, and it is of similar order even otherwise. So the probability of being *at* xo, which is surely less than the probability of being in the segment, must be less than 1/n for all n. It is therefore zero.
What is not zero is, indeed, the probability density (or intensity, as you say). But this is a very different kind of object. We know about probabilities, between zero and one and related to our belief in the possibility of an event happening in a certain way, but probability densities, which can take arbitrarily large values, and even diverge, are less obvious. So I suggest to look explicitly at the probability of being in a given interval. That relates, of course to the definition of probability density. My point is, we should be clear about the difference.
This may help to understand the original question: no, there is nothing in those points to prevent the presence of the particle. Rather, the state of the particle is of such a nature that it is quite unlikely to be in the vicinity of a node: indeed the probability of being within length l of a node decreses faster with l than the corresponding probability for a point that is not a node.
Good answer, Dr. Leyvraz! I begin to see clearly in this subject, which has haunted me for years. Thank you.
I guess the math has been discussed here, so let me just add something on the intuitive understanding why the particle "cannot be at one point" around the node. Maybe it helps to keep in mind that such stationary (energy eigen-)states are essentially standing waves. This is similar to an ordinary drum or membrane, where you see certain standing-wave patterns, see eg.
http://en.wikipedia.org/wiki/Normal_mode
The probability wave function may be visualized in an analogous manner.
Sascha,
You are right. The confusion should be over now. At the node there is a mutual cancellation of two oppositely directed waves, so to say.
Regards one and all,
Rajat
Prof. Leyvraz, why did you say that the probability of the particle being in the vicinity of a node decreases as l cubed? in a Taylor series expansión, I get l squared. (that is fast enough in comparison with l, however.)
This an interesting question, however, I am hesitant to make any physical interpretations regarding a single isolated point.
Some time ago this point (no pun intended) was discussed in another thread and the conclusions, if I remember correctly were that: "The concept of a mathematical point is unphysical and starting any fundamental description of physical reality with points and trajectories (although classically very successful) would nonetheless be severely limiting and not fundamental".
I think Prof. Leyvraz said about the same thing. In QM we have a probability density that concerns the environment of "the point" which permits more realistic interpretations.
@Raul: for a nonsingular wavefuncion psi(x) goes as x if there is a node, say, at 0. Then the probability density |psi(x)^2 goes as x^2, and the integral from 0 to l, which gives the probability, goes as l^3. Of course, you could have a node in which psi(x) does not go linearwly to 0, but it would be kind of singular.|
Leyraz
The question is on a particle in a 1-D box for which the results are continuous at x=0 (sinus-functions)
Again, thank you to everyone. I am beginning to see things more clearly.
Dear Raul,
the Schrödinger equation and the wave functions you refer to are non-relativistic. If you take relativistic wave equation, everything looks different. E.g., the Dirac wave function of an electron has nodes in its individual components, but the probability which depends on all four wave function components does not vanish locally. In the Klein-Gordon case, the localization of a particle is also more involved than in the nr case. The vanishing probability is a well-known artefact of first-quantized, non-relativistic theories.
Sincerely,
Andreas
Dear Raul,
Looking at so big and technical answers I am not sure that I am suitable person to answer this but still trying to explain simple things I know.
If we consider example of 1D particle in box node is simply a point where the wave and reflected wave are out of phase at all t (this can be easily shown by sinusoidal wave equation). So at that point the wave interfering is resulting destructive interference at all t causing intensity (better word will be power but intensity is more popular to use) always zero. Now if if we see look at it from particle point of view if this leads to number density of photon at that point zero which is obviously causing probability zero.
In simpler word we should not assume a bounded electron as a single wave but it is standing wave which is formed by interference between its own reflected waves by boundary (just like a vibrating guitar string). If we consider electron as free particle in space (I am not sure about this part) will lead to simple a travelling wave which will have no nodes and antinode, so uniform probability density. And also it will lead to energy of unbounded electron will not be quantized (this one I am sure ;) ).
In short at node all component wave cancel each of with out any physical (mysterious barrier). No one is stopping anything just two disturbance in opposite direction are cancelling each other. (If they are equal in magnitude it will lead to zero otherwise it can be non zero minima).
Hopefully this have something what you were looking for.
Regards,
Ankit anand
Schrodinger equation is a linear equation. An important consequence of the linearity property is superposition principle. This means that if:
ψ1 and ψ2 are two wave functions then ψ3=α1 ψ1 + α2 ψ2 is also a wave function. Also, one can show that if ψ1 and ψ2 are square integrable functions so is ψ3 in general. This point has been discussed above. However if ψ1 and ψ2 have nodes ψ3 will also have nodes in general. Therefore we come back again to the same issue.
At the point where |ψ3(x)|2 vanishes the particle will be absent at all times (this is because the time dependent part is a pure phase which disappears when you do mod squaring). In other places where it is non-zero the particle will be present. A similar situation is seen experimentally in interference experiments. Where we see dark spots in the interference pattern. I have a picture of Feynmann, explaining this on board. I hope you will like it.
Apparently, a node of psi_3 must be simultaneously a node of psi_1 and of psi_2.
Thank you, Biswajoy.
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