List of the most fundamental assumptions of quantum mechanics?

Can anybody give a concise list of the most fundamental assumptions of quantum mechanics in plain english ?

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what do you mean with assumptions ? – Stefano Borini Nov 10 '10 at 0:39
Just like we have a dimensionless point particle, the constant linear flow of time, and an immovable space fabric having straight lines in the classical Newtonian mechanics ? – mumtaz Nov 10 '10 at 0:45
Sadly, quantum mechanics is not written in plain English. – Mark Eichenlaub Nov 10 '10 at 1:57
@MarkE And there does not exist a continuous mapping of quantum mechanics into everyday language. – Mark C Nov 10 '10 at 6:04
as a quantum chemist would say, yes, it exists, but its projection in the english space has lower dimension, thus has a larger error due to the variational theorem. – Stefano Borini Nov 10 '10 at 8:39

Uncertain Principles: 7 essential elements of QM

Sorry but I can't be made plainer than that in English.

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 Especially well told in point #5 is that the nonlocal effect is seen only when the measurements are brought together in one place. That detail is too often overlooked by writers trying to convey quantum mechanics to non-physicists. – DarenW Aug 1 '11 at 7:25 I cannot agree with Orzel, mainly because he is adding interpretation to the principles, which muddies the waters too much to be an acceptable answer. – joseph f. johnson Jan 14 '12 at 16:59

In a rather concise manner, Shankar describes four postulates of nonrelativistic quantum mechanics.

I. The state of the particle is represented by a vector $|\Psi(t)\rangle$ in a Hilbert space.

II. The independent variables $x$ and $p$ of classical mechanics are represented by Hermitian operators $X$ and $P$ with the following matrix elements in the eigenbasis of $X$

$$\langle x|X|x'\rangle = x \delta(x-x')$$

$$\langle x|P|x' \rangle = -i\hbar \delta^{'}(x-x')$$

The operators corresponding to dependent variable $\omega(x,p)$ are given Hermitian operators

$$\Omega(X,P)=\omega(x\rightarrow X,p \rightarrow P)$$

III. If the particle is in a state $|\Psi\rangle$, measurement of the variable (corresponding to) $\Omega$ will yield one of the eigenvalues $\omega$ with probability $P(\omega)\propto |\langle \omega|\Psi \rangle|^{2}$. The state of the system will change from $|\Psi \rangle$ to $|\omega \rangle$ as a result of the measurement.

IV. The state vector $|\Psi(t) \rangle$ obeys the Schrödinger equation

$$i\hbar \frac{d}{dt}|\Psi(t)\rangle=H|\Psi(t)\rangle$$

where $H(X,P)=\mathscr{H}(x\rightarrow X, p\rightarrow P)$ is the quantum Hamiltonian operator and $\mathscr{H}$ is the Hamiltonian for the corresponding classical problem.

After that, Shankar discusses the postulates and the differences between quantum mechanics and classical mechanics, hopefully in plain english. Probably you want to take a look at that book: Principles of Quantum Mechanics

There are, of course, other sets of equivalent postulates.

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 +1, I was thinking of the same thing, I just couldn't remember the list of assumptions in detail. – David Zaslavsky♦ Nov 10 '10 at 2:26 Ah, the first mention of Shankar I've seen! – Mark C Nov 10 '10 at 6:04 Thanks for the conciseness and reference to Shankar's book. I will definitely have a go at it. – mumtaz Nov 10 '10 at 8:27 This is hardly in plain english ;) – Stefano Borini Nov 10 '10 at 8:28 @Stefano ...yeah its not , thats why i up-voted it but could not mark it as the answer ;) – mumtaz Nov 10 '10 at 8:43
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Here are some ways of looking at it for an absolute beginner. It is only a start. It is not the way you will finally look at it, but there is nothing actually incorrect here.

First, plain English just won't do, you are going to need some math. Do you know anything about complex numbers? If not, learn a little more before you begin.

Second, in quantum mechanics, we don't work with probabilities, you know, numbers between 0 and 1 that represent the likelihood of something happening. Instead we work with the square root of probabilities, called "probability amplitudes", which are complex numbers. When we are done, we convert these back to probabilities. This is a fundamental change; in a way the complex number amplitudes are just as important as the probabilities. The logic is therefore different from what you are used to.

Third, in quantum mechanics, we don't imagine that we know all about objects in between measurements, rather we know that we don't. Therefore we only focus on what we can actually measure, and possible results of a particular measurement. This is because some of the assumed imagined properties of objects in between measurements contradict one another, but we can get a theory where we can always work with the measurements themselves in a consistent way. Any measurement (meter reading, etc.) or property in between measurements has only a probability amplitude of yielding a given result, and the property only co-arises with the measurement itself.

Fourth, if we have several mutually exclusive possibilities, the probability amplitude of either one or the other happening is just the sum of the probability amplitudes of each happening. Note that this does not mean that the probabilities themselves add, the way we are used to classically. This is new and different.

If we add up the probabilities (not the amplitudes this time but their "squares") of all possible mutually exclusive measurements, gotten by "squaring" the probability amplitudes of each mutually exclusive possibilities, we always get 1 as usual. This is called unitarity.

This is not the whole picture, but it should help somewhat. As a reference, try looking at three videos of some lectures that Feynman once gave at Cornell. That should help more.

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 thanks for the link to vids – mumtaz Nov 10 '10 at 8:52 The probability amplitudes are not quite « square roots » of the probabilities, they differ from the square root by a phase factor which is important. Perhaps a parenthetical remark should be put into the third paragraph to this effect. – joseph f. johnson Jan 14 '12 at 17:07 @josephf.johnson: One can declare them to be signed square roots if one implements "operator i" and doubles the size of the Hilbert space. This is a clearer point of view for those who do not like complex numbers sneaking into physics without a physical argument. – Ron Maimon Apr 27 '12 at 18:22 The suggestion you sketch here will indeed work, but doesn't change the issue I brought up, of the phase factors. Relative phase factors are physical, whether you use the formalism of complex numbers or use an alternative real space with double the dimension (this procedure is called base change). – joseph f. johnson Apr 28 '12 at 15:36

I don't really understand exactly what you mean with assumptions, but I guess you ask about the physical nature and behavior of the quantistic world. I am going to be very inaccurate in the following writing, but I need some space to get things understood. What you are going to read is a harsh simplification.

The first assumption is the fact that particles at the quantistic level don't have a position, like real-life objects, they don't even have a speed. Well, they sort of have it, but in reality they are "fuzzy". Fuzzy in the sense that you cannot really assign a position to them, or a speed.

To be fair, you perfectly know a real-world case scenario where you have to give up the concepts of speed and position for an entity. Take a guitar, and pluck a string. Where is the "vibration" ? Can you point at one specific point in the string where the "vibration" is, or how fast this vibration is traveling. I guess you would say that the vibration is "on the string" or "in the space between one end of the string, and the other hand". However, there are parts of the string that are vibrating more, and parts that are not vibrating at all, so you could rightfully say that a larger part of the "vibration" is in the center of the string, a smaller part is on the lateral parts close to the ends, and no vibration is at the very ends. In some sense, you will experience higher "presence" of the vibration in the center.

With electrons, it's exactly the same. Think "electron" as "vibration" in the previous example. Soon, you will realize that you cannot really claim anything about the position of an electron, for the fact that has wave nature. You are forced to see the electron not as a charged ball rolling here and there, but as a diffuse entity totally similar to the concept of "vibration". As a result, you cannot say anything about the position of an electron, but you can claim that there are zones in the space where the electron has "higher presence" and zones that have "smaller presence". These zones are probabilistic in nature, and this probability is directly obtained by a mathematical description called the wavefunction.

The wavefunction mainly depends on the external potential, meaning that the presence of charges, such as protons and other electrons, will affect the perceived potential and will have an effect on the probability distribution in space of an electron.

A second assumption is the mapping between physical quantities (such as energy, or momentum) and quantistic operators. Take for example the momentum of an everyday object: it is given by its mass times its speed. In the quantistic world, you don't really have the concept of position, hence you don't have the concept of speed, hence you have to reinterpret the whole thing in terms of probability (rememeber the string ?), which means that what you have about a quantistic object is its wavefunction, and you have to extract the information about the momentum from this wavefunction. How do you do it ?

Well, there's a dogma in quantum mechanics, that if you apply a magic "operator" to the wavefunction, it gives you the quantity you want. There are (simple) rules to generate these operators, and you can apply them to a wavefunction to query the momentum, or the total energy, the kinetic energy and so on.

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so the first thing we notice down there is a clear shift from a deterministic to a probabilistic notion of existence ? – mumtaz Nov 10 '10 at 0:57
I would not say "probabilistic existence". It's not like if you say "the electron has, say, 40% probability of being here" does not mean that it spends its time 40% here and 60% somewhere else. I think the plucked string is the most appropriate real-world example. It's not like in the middle of the string there's all the vibration 40 % of the time, and 60% is somewhere else. The vibration is just 40 % there and 60 % on the rest of the string. – Stefano Borini Nov 10 '10 at 1:05
so we dont have real dimensionless point particles then. These are extended in space , right ? ...but wait are we talking about the same space that we have an intuitive notion of from our empirical experience or is the space down there also funky ? – mumtaz Nov 10 '10 at 1:21
@mumtaz:The electron is technically handled as a dimensionless point, but it's irrelevant. What you work on is the coordinate of an electron, but what you have is a coordinate of space where an electron's wavefunction has a given value. when you say wavefunction $\psi(x)$, your $x$ is a dimensionless point, but it's not really like you have a infinitesimal particle stuck in point $x$. – Stefano Borini Nov 10 '10 at 8:36

Dirac seems to have thought that the most fundamental assumption of Quantum Mechanics is the Principle of Superposition I read somewhere, perhaps an obituary, that he began every year his lecture course on QM by taking a piece of chalk, placing it on the table, and saying, now I paraphrase:

One possible state of the piece of chalk is that it is here. Another possible state is that it is over there (pointing to another table). Now according to Quantum Mechanics, there is another possible state of the chalk in which it is partly here, and partly there.

In his book he explains more about what this « partly » means: it means that the properties of a piece of chalk that is partly in state 1 (here) and partly in state 2 (there), are in between the properties of state 1 and state 2.

EDIT: I find I have compiled the basic explanations given by Dirac in the first edition of his book. Unfortunately, it is relativistic, but I am not going to re-type it all.

« we must consider the photons as being controlled by waves, in some way which cannot be understood from the point of view of ordinary mechanics. This intimate connexion between waves and particles is of very great generality in the new quantum mechanics. ...

« The waves and particles should be regarded as two abstractions which are useful for describing the same physical reality. One must not picture this reality as containing both the waves and particles together and try to construct a mechanism, acting according to classical laws, which shall correctly describe their connexion and account for the motion of the particles.

« Corresponding to the case of the photon, which we say is in a given state of polarizationn when it has been passed through suitable polarizing apparatus, we say that any atomic system is in a given state when it has been prepared in a given way, which may be repeated arbitrarily at will. The method of preparation may then be taken as the specification of the state. The state of a system in the general case then includes any information that may be known about its position in space from the way in which it was prepared, as well as any information about its internal condition.

« We must now imagine the states of any system to be related in such a way that whenever the system is definitely in one state, we can equally well consider it as being partly in each of two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannnot be conceived on classical ideas.

...

« When a state is formed by the superposition of two other states, it will have properties that are in a certain way intermediate between those of the two original states and that approach more or less closely to those of either of them according to the greater or less weight' attached to this state in the superposition process.

« We must regard the state of a system as referring to its condition throughout an indefinite period of time and not to its condition at a particular time, ... A system, when once prepared in a given state, remains in that state so long as it remains undisturbed.....It is sometimes purely a matter of convenience whether we are to regard a system as being disturbed by a certain outside influence, so that its state gets changed, or whether we are to regard the outside influence as forming a part of and coming in the definition of the system, so that with the inclusion of the effects of this influence it is still merely running through its course in one particular state. There are, however, two cases when we are in general obliged to consider the disturbance as causing a change in state of the system, namely, when the disturbance is an observation and when it consists in preparing the system so as to be in a given state.

« With the new space-time meaning of a state we need a corresponding space-time meaning of an observation. [, Dirac has not actually mentioned anything about observation' up to this point...]

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