Quantum Mechanics without Hilbert Space

[Chopin]

When I asked this question "Why did you mention about the coulomb potential? Are you giving it as example that it can only be modeled by Hilbert space and not by fourier?" I was only thinking in terms of Sense1. But you didn't reply yes or no. So I thought you were trying to say Fourier is still somehow used in coulomb potential in some way. But now knowing the context that I only meant Sense1. Then your answer is "yes" to the question "Are you giving it as example that it can only be modeled by Hilbert space and not by fourier?", right? Look. English is not my native language, and since I'm a novice in QM. I need a yes or no answer to be definite of my question. So just answer "right" if you are saying that the coulomb potential can only be modeled by Hilbert space and not by Fourier which I only understand use Sense1 which involve sine waves. This is to be 100% sure of your answer before I leave this thread.

No, the solutions to the Coulomb potential cannot be modeled by a Fourier series, since the component solutions aren't sine waves.

In the infinite square well, the solutions are a set of sine waves, so combining them together forms a Fourier series. For the Coulomb potential, the solutions are complicated Bessel functions. You can still add them together, but since they aren't sine waves, it isn't really a Fourier series anymore.

More importantly, though, "Hilbert space" and "Fourier" aren't the same kind of thing, so you can't even really compare them. "Hilbert space" and "wavefunction" are the same kind of thing. Specifically, the wavefunction is just a shorthand for talking about the position basis of the Hilbert space.

Hmm... so the words "wave function" always involve the position basis. Can you please give example of QM application that doesn't use the position basis and hence Hilbert Space is used. Maybe coulomb potential is one? What else to get the idea. Now if you no longer use wave function but Hilbert space. What is the proper term for that. For example. What is the replacement for the term "Solve for the wave function"... maybe "solve for the Hilbert space"?, this doesn't seem right. What is the right term to use? Thanks.

The best example of problems that don't involve the position basis are operators with discrete eigenvectors. An easy example is photon polarization. A beam of photons can either be polarized horizontally or vertically. So to describe a photon's polarization, you need a two-dimensional Hilbert space. We can denote a photon that's polarized vertically as |V\rangle, and one that's polarized horizontally as |H\rangle. But a photon can also be in a superposition of horizontal and vertical. We denote this as |V\rangle + |H\rangle.

All of these states are vectors in a two-dimensional Hilbert space that describes the range of polarizations that a photon can have. This sort of problem is impossible to describe using a wavefunction, because there's no way to describe the problem in the position basis. A good text that describes these sorts of problems is the Feynman Lectures--he spends several chapters talking about discrete-valued problems like polarization and spin, and only then moves on to continuous operators like position and momentum.

 

[Varon]

This is not quite true. In fact, you already saw the definition in the Wikipedia page that a wave function refers to an element of a complex Hilbert space. For example, there is also a momentum space wave function, which is related to the position space wave function by the Fourier transform.

Chopin, hope you can comment on the above that it is more correct to refer to the wave function as an element of a complex Hilbert Space that is not necessarily position. You said: " "Hilbert space" and "wavefunction" are the same kind of thing. Specifically, the wavefunction is just a shorthand for talking about the position basis of the Hilbert space." But Truecrimson disagreed and said the wave function simply refers to an element of a complex Hilbert Space that is not necessarily position". Who is correct?

http://en.wikipedia.org/wiki/Momentum_space

You should really think of a wave function as an element in a Hilbert space (a vector space). How you actually write it down when, say, you want to integrate it, ultimately depends on your choice of basis. This is exactly the same as writing a vector as a column of numbers. If the basis is not known, there is no way to know what those number mean. And, of course, you have the freedom to choose a basis that simplifies your problem. For example, the functional form of the Hamiltonian operator depends on the basis. In particular, the momentum operator in the momentum space is just a multiplication by momentum. So for a case of a free particle, changing to the momentum basis reduces the Schroedinger equation from a second order PDE to a first order ODE.

 

[Chopin]

Chopin, hope you can comment on the above that it is more correct to refer to the wave function as an element of a complex Hilbert Space that is not necessarily position. You said: " "Hilbert space" and "wavefunction" are the same kind of thing. Specifically, the wavefunction is just a shorthand for talking about the position basis of the Hilbert space." But Truecrimson disagreed and said the wave function simply refers to an element of a complex Hilbert Space that is not necessarily position". Who is correct?

The wavefunction is almost always used as a way of describing the state in position basis. You could transform it into a wavefunction over momentum basis if you wanted to, though, because those two observables are both continuous. They're just two different bases for the Hilbert space, so the two different wavefunctions are just different representations of the same state. It's like if you have a vector, and then view it from a rotated coordinate system--the vector hasn't changed at all, but the numbers that describe it do, because the coordinate system is different.

What you can't do, though, is use a wavefunction to describe a particle with polarization or spin, because that observable is discrete.

 

[Varon]

The wavefunction is almost always used as a way of describing the state in position basis. You could transform it into a wavefunction over momentum basis if you wanted to, though, because those two observables are both continuous. They're just two different bases for the Hilbert space, so the two different wavefunctions are just different representations of the same state. It's like if you have a vector, and then view it from a rotated coordinate system--the vector hasn't changed at all, but the numbers that describe it do, because the coordinate system is different.

What you can't do, though, is use a wavefunction to describe a particle with polarization or spin, because that observable is discrete.

Ok. Thanks for all help. Are you a professor or student of physics?
Anyway. In the cosmology forum, I posted a question how much dynamics can occur in a superposition and a guy called Chalnoth answered: (what oscillations is he talking about?):

Chalnoth wrote:

"A large, complex object like a rock can't really be in a coherent superposition, let alone a galaxy.

Basically, the way we know about objects in coherent superpositions is through oscillation: we can observe the results of an object oscillating through, for instance, interference effects. But complex wavefunctions have oscillation times that tend to be very long, often much longer than the age of the universe.

And when your oscillation time is that long, there just isn't any way for the different components of the same wavefunction to obtain any information about one another. In fact, the different components of the wavefunction, when they are complex enough, interact so weakly with one another that they might as well be in different universes.

So anything as large as a galaxy in a superposition of states will behave exactly as if there was no superposition at all."

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

How true? I wanted to understand how much dynamics can exist in a superposition like whether a universe can be in superposition (in one branch) and evolve planets without having to use the branching in Many Worlds. Or maybe we can treat superposition as Potentia where there is potential to evolve and a realm of possibilities. But I can't understand how possibilities can suddenly give rise to planets... so maybe macroscopic superposition can only happen in Many Worlds and impossible as actually happening in real time in an object in one branch, isn't it. Supposed it could happen, how much dynamics can occur in a macroscopic superposition in one world I wonder. None according to Chalnoth, do you fully agree? And what is the oscillation thing he was describing?

 

[Chopin]

The oscillation thing refers to the fact that any definite-energy solution to the Schrodinger Equation looks like this:

\Psi(\textbf{x},t) = e^{iEt/\hbar}F(\textbf{x})

That is, it has a spatially stationary solution, but the entire thing rotates through the complex plane at a frequency defined by the energy E. If the system is in a superposition of these states, the various base states rotate at different frequencies, and the changing interference effects that come out of this leads to a spatially-changing waveform (think about beat frequencies in classical wave theory.)

I don't exactly know what all of the superposition stuff in that post is about. You'd probably be best off making a new post for that, since it's not really related to the original topic of this post.

 

[Varon]

The oscillation thing refers to the fact that any definite-energy solution to the Schrodinger Equation looks like this:

\Psi(\textbf{x},t) = e^{iEt/\hbar}F(\textbf{x})

That is, it has a spatially stationary solution, but the entire thing rotates through the complex plane at a frequency defined by the energy E. If the system is in a superposition of these states, the various base states rotate at different frequencies, and the changing interference effects that come out of this leads to a spatially-changing waveform (think about beat frequencies in classical wave theory.)

I don't exactly know what all of the superposition stuff in that post is about. You'd probably be best off making a new post for that, since it's not really related to the original topic of this post.

Actually I had made a new post about it last May 3 here in Quantum forum with the title "Oscillations in Wave Function", but only one replied and he was not sure and didn't follow up:

https://www.physicsforums.com/showthread.php?t=495770

The original thread was in Cosmology forum called "Superposition and Big Bang"

https://www.physicsforums.com/showthread.php?t=495442

But the guy who replied believed in Many Worlds and didn't like Copenhagen where Superpositions occur in one world only. So I wonder if his reply which I quoted to you above is due to bias. Hence I'm repeating it here for independent verification by one who truly knows about QM like you and unbiased about interpretations. So you don't agree too that complex dynamics can occur inside a superposition? The context of what I meant was this I posted in the original Cosmology forum:

"Anyone familiar with both quantum mechanics and cosmology here. Say. How much dynamics can occur in the deterministic Schroedinger Equation in its evolution while it is in unitary state (before collapse)? For example. Is it possible for the Big Bang and evolution of stars to solar system and planets to occur while everything is inside a superposition (that doesn't involve Many Worlds but just as superposition of possibilities)? I'm trying to analyze the physicist Wigner who proposed that consciousness caused collapse of the wave function. Before life begins on earth. He seems to be saying that the universe is in a state of superposition. After life reaches a certain theshold. It finally collapsed the wave function of the Earth and the surrounding. Is this possible? Does the Schroedinger Equations allows for instance the evolution of a solar system while it is still unitary and before the wave function collapse? Or is it not possible? "

Well? (this question would be the last in this thread... thanks for every help :) )

 

[Fredrik]

Varon, you might want to look into the phenomenon called "decoherence". The Schrödinger equation describes how the state of a system that's completely isolated from its environment changes with time, but as soon as the system interacts with its environment, the interaction will very quickly turn a superposition into a state that's indistinguishable from a classical probability distribution. (That's what's called decoherence). For example, if you put a particle in a superposition of two localized states, let's say |left>+|right>, interactions with air molecules will often be sufficient to very quickly change the state into one that's very close to "either |left> or |right>, with equal probabilities for both".

It's quite hard to prevent a superposition from "decohering" into a classical probability distribution. It can be hard even with atoms and molecules. It will probably be forever beyond our technological capabilities to do it with something larger, say a bacterium. It just isn't possible to keep it sufficiently isolated from its environment. I think even the fact that different parts of it are interacting with the other parts will mess things up.

Now, if we can't even keep buckyballs in superpositions when they interact with the surrounding air, imagine how hard it would be for galaxies to stay in superpositions when they interact gravitationally. I think quantum effects are just unimaginably small at those scales, and I don't just mean that they would be 10 times smaller than what can be detected, or even a googolplex times smaller. I don't know what the correct number is, but I'm pretty sure it's bigger than that.

 

[Varon]

Varon, you might want to look into the phenomenon called "decoherence". The Schrödinger equation describes how the state of a system that's completely isolated from its environment changes with time, but as soon as the system interacts with its environment, the interaction will very quickly turn a superposition into a state that's indistinguishable from a classical probability distribution. (That's what's called decoherence). For example, if you put a particle in a superposition of two localized states, let's say |left>+|right>, interactions with air molecules will often be sufficient to very quickly change the state into one that's very close to "either |left> or |right>, with equal probabilities for both".

It's quite hard to prevent a superposition from "decohering" into a classical probability distribution. It can be hard even with atoms and molecules. It will probably be forever beyond our technological capabilities to do it with something larger, say a bacterium. It just isn't possible to keep it sufficiently isolated from its environment. I think even the fact that different parts of it are interacting with the other parts will mess things up.

Now, if we can't even keep buckyballs in superpositions when they interact with the surrounding air, imagine how hard it would be for galaxies to stay in superpositions when they interact gravitationally. I think quantum effects are just unimaginably small at those scales, and I don't just mean that they would be 10 times smaller than what can be detected, or even a googolplex times smaller. I don't know what the correct number is, but I'm pretty sure it's bigger than that.

I'm talking about the Universe as a whole that is a closed system and isolated and no decoherence occurs from outside. Wigner said consciousness can cause collapse. So before the universe evolved conscious beings, the universe was in superposition.. but only in one world or branch (here let's avoid Many Worlds for sake of discussion). After evolution reached a threshold, the Earth wave function collapsed and it spread to the entire universe. So my question is. Before humans developed. Could a fish evolve into a dolphin and larger beings inside a superposition that doesn't involve Many worlds. This is the reason why I asked if dynamics can occur inside a superposition in one world and to what extend. Chalnoth stated that oscillations prevented that. But since he is a biased Many Worlds believer. I asked pure Copenhagenist like Chopin for second opinion. That's all there is to it.

Anyway. Since you have mentioned about Decoherence. I'm quite familiar about Decoherence, but there is one thing that bothered me for weeks. Supposed there was no decoherence (remember the word "supposed" meaning for theoretical understanding), and classical states were not chosen as preferred basis. And the world would become entirely quantum even in big macroscopic objects. Can we now said to perceive the superpositions? What confused me is that the wave function is supposed to be only knowledge of the observer. So without decoherence and without observers. Would the macroscopic world be in quantum superposition that is ontological reality.. meaning that actually happens?

Bottom line is. Supposed there was no decoherence ("supposed), and no observers and the world didn't have classical reality. What how would be the ontological form of the world look like.. as a macroscopic quantum superposition (in one world.. let's avoid Many worlds for now for this theoretical understanding)?? This is all I need to know so no need for separate thread. Thanks.

 

[Chopin]

I asked pure Copenhagenist like Chopin for second opinion.

I'm not sure where I gave the impression that I was a believer in the Copenhagen interpretation. For the most part, I don't really care which interpretation is "right", since so far there doesn't seem to be any way to tell the difference between them (I guess that basically means I'm a believer in Feynman's "shut up and calculate" interpretation...) But I find the notion of an explicit and physical wavefunction collapse to be as distasteful as the next guy, and the decoherence interpretations (many-worlds, consistent histories, etc.) have an elegance to them that appeals to me, although I don't know nearly enough about them yet to really make an educated decision one way or the other.

Bottom line, though, is that for the types of questions you're asking, the interpretation doesn't matter--the math always does exactly the same thing. So if you're specifically interested in understanding things about the Hilbert space formalism, wavefunctions, operators, etc. then it makes absolutely no difference which interpretation you subscribe to. Any experiment that we can conceivably perform which is described by quantum mechanics will give exactly the same results regardless of which interpretation you use, so the question of which one is "right" is largely a metaphysical one.

 

[Varon]

The oscillation thing refers to the fact that any definite-energy solution to the Schrodinger Equation looks like this:

\Psi(\textbf{x},t) = e^{iEt/\hbar}F(\textbf{x})

That is, it has a spatially stationary solution, but the entire thing rotates through the complex plane at a frequency defined by the energy E. If the system is in a superposition of these states, the various base states rotate at different frequencies, and the changing interference effects that come out of this leads to a spatially-changing waveform (think about beat frequencies in classical wave theory.)

I don't exactly know what all of the superposition stuff in that post is about. You'd probably be best off making a new post for that, since it's not really related to the original topic of this post.

In the above. Are you saying that quantum superposition involves only "spatially stationary solution" meaning there is not much movement or dynamics occurring inside. Are you then confirming that complex dynamics can't exist inside a superposition? You didn't state this directly. Since i can't read between the lines because english is not my native language and I'm not very good in reading. I'm not 100% sure you meant it. So pls. confirm and this wraps up this thread. Many thanks.

 

[Chopin]

In the above. Are you saying that quantum superposition involves only "spatially stationary solution" meaning there is not much movement or dynamics occurring inside. Are you then confirming that complex dynamics can't exist inside a superposition? You didn't state this directly. Since i can't read between the lines because english is not my native language and I'm not very good in reading. I'm not 100% sure you meant it. So pls. confirm and this wraps up this thread. Many thanks.

No, it means only that the equation can be separated into a time-dependent part and a space-dependent part. An example would be a good old standing wave, like this:

\Psi(x,t) = e^{iEt/\hbar}sin(2\pi x/k)

This equation will have nodes along the x-axis with a wavelength of k, and the complex phase will rotate around with a frequency of E. Because the position of the nodes doesn't change, though, the wave stays in the same spot in space (i.e. it is a standing wave, not a traveling wave.)

However, imagine superposing two of these waves together, with different periods in space, and different frequencies in time. The waves will interfere with each other, just like classical waves, and so the result won't have nodes that stay in place anymore--the waveform will look like it's moving around. That's how the oscillations can produce dynamics--even though each single solution might be stationary in space, the interference of multiple solutions in a superposition can lead to a waveform which moves in time.

 

[Varon]

No, it means only that the equation can be separated into a time-dependent part and a space-dependent part. An example would be a good old standing wave, like this:

\Psi(x,t) = e^{iEt/\hbar}sin(2\pi x/k)

This equation will have nodes along the x-axis with a wavelength of k, and the complex phase will rotate around with a frequency of E. Because the position of the nodes doesn't change, though, the wave stays in the same spot in space (i.e. it is a standing wave, not a traveling wave.)

However, imagine superposing two of these waves together, with different periods in space, and different frequencies in time. The waves will interfere with each other, just like classical waves, and so the result won't have nodes that stay in place anymore--the waveform will look like it's moving around. That's how the oscillations can produce dynamics--even though each single solution might be stationary in space, the interference of multiple solutions in a superposition can lead to a waveform which moves in time.

Ows. Ok. I wonder why you are not a Science Advisor at PhysicsForum. You explained even better than Neumaier.

Anyway. I was studying the history of the concept of superposition. When Schroedinger first developed the equation and thought the wave was some kind of charge density or mass density. This concept didn't allow superposition of say going to left or right slit in double slit, isn't it? Did people realize the application of the possibilities of superposition (of different outcomes) only occurs after Born made the realization of the probabilities interpretation of the wave amplitude squared or was it after the Hilbert space was introduced by Dirac?

 

[Chopin]

When Schroedinger first developed the equation and thought the wave was some kind of charge density or mass density. This concept didn't allow superposition of say going to left or right slit in double slit, isn't it? Did people realize the application of the possibilities of superposition (of different outcomes) only occurs after Born made the realization of the probabilities interpretation of the wave amplitude squared or was it after the Hilbert space was introduced by Dirac?

I don't quite know the history on how this developed, but you are correct--the wavefunction as a charge density can't act like a superposition. It can't describe the notion of finding the particle at a precise position, so it can only act like a wave.

The wavefunction under the probability interpretation, though, does allow for superposition. If \Psi is nonzero over a volume of space, it means the particle is in a superposition of being in all places inside the volume. The Hilbert space concept simply extends this idea, and provides a way to talk about superposition in other contexts (like polarization, spin, etc.)

 

[Varon]

I don't quite know the history on how this developed, but you are correct--the wavefunction as a charge density can't act like a superposition. It can't describe the notion of finding the particle at a precise position, so it can only act like a wave.

The wavefunction under the probability interpretation, though, does allow for superposition. If \Psi is nonzero over a volume of space, it means the particle is in a superposition of being in all places inside the volume. The Hilbert space concept simply extends this idea, and provides a way to talk about superposition in other contexts (like polarization, spin, etc.)

As a pragmatist who focuses only in the mathematical sense (shut up and calculate approached you mentioned), what do you really think happen to the 430-atom buckyball inside the double slit as it is emitted and detected by the detector with inteferences forming even if one buckyball is sent one at a time. The wavefunction is only our knowledge of the object, isn't it. The wave function is not the object itself. Or is it? If it is the object itself, it can morph into wave and interfere at the slits and then morph into particle when it reaches the screen. You can't say that a quantum object is a wavicle and behave like particle or wave. Bigger object like a 430 atom buckyball is not tiny. It is big and pointless to call it a wavecle because we know it is a particle because it is a molecule. Perhaps back in Bohr times when people thought only electron can be quantum object, you can call electron a wavicle.. but not a 430 atom buckyball with many layers of information. Now if the wave function is only our knowledge of the experimental setup. What happens between to the buckyball between the emitter and the detector. What is your best guess? I still can't decide after years of thinking about it. Sometimes I like buckyball to be the wave function itself that can morph between particle and wave, but Wigner Friend experiment seems to suggest the wave function can't be in the object or else we would observe different outcomes. But a guy called Fra believes this is so and observers can see different things simultaneously. Weird. So what is your best guess? Pick one, don't say no and don't care (shut up and calculate). Because the unification of quantum mechanics and general relativity may require us to understand the measurement problem. This is the main reason I'm interested in all this.. to solve for Quantum Gravity.

 

[alxm]

I was studying the history of the concept of superposition. When Schroedinger first developed the equation and thought the wave was some kind of charge density or mass density. This concept didn't allow superposition of say going to left or right slit in double slit, isn't it? Did people realize the application of the possibilities of superposition (of different outcomes) only occurs after Born made the realization of the probabilities interpretation of the wave amplitude squared or was it after the Hilbert space was introduced by Dirac?

I'm not sure that's quite right. I'm pretty sure Schrödinger knew it was the absolute square of the wave function that gave the charge density. Besides that you'd have a problem with the complex values, he wouldn't have been able to arrive at the well-known justification for the Bohr radius otherwise. Superpositions are somewhat implicit in the math, although I don't think Schrödinger had a very clear idea at the start about them, or what the phase meant.

Heisenberg and Born were concurrently working on Matrix mechanics, which did explicitly deal with probabilities, and Dirac famously showed the two were equivalent. I think it was von Neumann who introduced the Hilbert space concept though.

 

[qsa]

As a pragmatist who focuses only in the mathematical sense (shut up and calculate approached you mentioned), what do you really think happen to the 430-atom buckyball inside the double slit as it is emitted and detected by the detector with inteferences forming even if one buckyball is sent one at a time. The wavefunction is only our knowledge of the object, isn't it. The wave function is not the object itself. Or is it? If it is the object itself, it can morph into wave and interfere at the slits and then morph into particle when it reaches the screen. You can't say that a quantum object is a wavicle and behave like particle or wave. Bigger object like a 430 atom buckyball is not tiny. It is big and pointless to call it a wavecle because we know it is a particle because it is a molecule. Perhaps back in Bohr times when people thought only electron can be quantum object, you can call electron a wavicle.. but not a 430 atom buckyball with many layers of information. Now if the wave function is only our knowledge of the experimental setup. What happens between to the buckyball between the emitter and the detector. What is your best guess? I still can't decide after years of thinking about it. Sometimes I like buckyball to be the wave function itself that can morph between particle and wave, but Wigner Friend experiment seems to suggest the wave function can't be in the object or else we would observe different outcomes. But a guy called Fra believes this is so and observers can see different things simultaneously. Weird. So what is your best guess? Pick one, don't say no and don't care (shut up and calculate). Because the unification of quantum mechanics and general relativity may require us to understand the measurement problem. This is the main reason I'm interested in all this.. to solve for Quantum Gravity.

Varon, I have sent you a private message you can read it by clicking on the message button on the upper right of PF screen. you will see that what Fra and Neumaier saying are not that strange after all, and ties a lot of the your questions.

 

[Fredrik]

I'm talking about the Universe as a whole that is a closed system and isolated and no decoherence occurs from outside.

I don't think you can assume that QM can be applied to the whole universe (or even to a physical system that has subsystems that we would experience as behaving classically) without turning it into a many-worlds theory. You said that you didn't want a many-worlds answer, so I'm not sure how I should respond.

The reason that many worlds enter the picture automatically is that we know that a person's experiences are described by one of the terms in a state operator (=density matrix), but there's nothing in the theory that gives one of the terms a different meaning than the others. The most straightforward interpretation of this is that every term describes something that's actually happening. (If only one of them does, then there has to be an unknown mechanism that singles out one of the terms as "special". This would imply that the theory needs to be modified, while the many-worlds interpretation doesn't).

Wigner said consciousness can cause collapse.

Wigner and von Neumann, probably the two greatest mathematical physicists of that time, both speculated along those lines. But that doesn't make it right. It's still just a wild speculation from a time when QM was less understood than it is today. I think it's one of the worst ideas ever introduced into physics to be honest.

I'm quite familiar about Decoherence, but there is one thing that bothered me for weeks. Supposed there was no decoherence (remember the word "supposed" meaning for theoretical understanding)

I have no problem with hypothetical scenarios that involve unrealistic assumptions, but to suppose that there's no decoherence is to suppose that we're talking about a world where QM can't even make good predictions about results of experiments. So what theory should I use to answer the question? Perhaps you just meant "consider a system whose interactions with the environment are negligible for some time t". This assumption is OK, since it doesn't contradict QM. In a world without gravity, the system could probably be quite large.

and classical states were not chosen as preferred basis.

The "preferred" basis is determined by the interaction between the system and its environment. So this part of what you're supposing seems to contradict QM too.

Let me try to fix those assumptions for you. Suppose that a physicist and his laboratory existed in a world without gravity and without any other matter. The physicst sets up and performs a QM experiment that has a 50% chance of amputating his legs. Will he now experience a superposition of having legs and not having legs? A "Copenhagenish" answer is that this situation is no different from when you perform an experiment in the lab. The theory tells you the probabilities of each possible result, nothing else. You have never experienced a superposition before, so why would this guy?

Bottom line is. Supposed there was no decoherence ("supposed), and no observers and the world didn't have classical reality. What how would be the ontological form of the world look like..

If QM applies to the whole universe, and we completely ignore the concept of "observer", there wouldn't be anything left to talk about other than the time evolution of the state of the universe. The state is always a unit vector in an infinite-dimensional Hilbert space, so the dynamics is described by a curve on the unit sphere of that space.

 

[gpran]

I have been reading the forum for some time as a new student of physics. My question is: Is the Hilbert Space different from the normal space? Why there is a different set of rules?

 

[Fredrik]

I have been reading the forum for some time as a new student of physics. My question is: Is the Hilbert Space different from the normal space? Why there is a different set of rules?

Have you studied linear algebra yet? A Hilbert space is an inner product space that satisfies one additional property (completeness) that makes it easier to work with. The "normal" space ℝ³ is a Hilbert space over the real numbers. (That last part means that you can multiply any member of ℝ³ with a real number and get a member of ℝ³ as the result). It's 3-dimensional, meaning that there exists a linearly independent set with 3 members, but no linearly independent set with 4 members.

The Hilbert space used in QM is over the complex numbers, and is infinite-dimensional. Its members represent possible states of a physical system.

 

[yoda jedi]

Von Neumann developed the concept of Hilbert Space in Quantum Mechanics. Supposed he didn't introduce it and we didn't use Hilbert Space now. What are its counterpart in pure Schroedinger Equation in one to one mapping comparison?

In details. I know that "the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators, and measurements are orthogonal projections" But this concept was developed by Von Neumann. Before he developed Hilbert Space. What are their counterpart in the pure Schroedinger equation up to Born interpretation of the amplitude square as the probability that electron can be found there?

Please answer more in words or conceptual and not with dense mathematical equations. Thanks.

Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space.
http://www.bbk.ac.uk/tpru/BasilHiley/Algebraic Quantum Mechanic 5.pdf

...These results have only intensified my curiosity as to why most if not all of the results
can be obtained without seemingly the need to resort to Hilbert space. This goes against
the prevailing orthodoxy that appears to insist that quantum mechanics cannot be done
except in the context of a Hilbert space. Yet there have been other voices raised against
the necessity of Hilbert space. Von Neumann himself wrote to Birkoff (1966) writing "I
would like to make a confession which may seem immoral: I do not believe absolutely in
Hilbert space any more." (A detailed discussion of why von Neumann made this
comment can be found in Rédei 1996).
But there are more important reasons why an algebraic approach has advantages. As
Dirac (1965) has stressed, when algebraic methods are used for systems with an infinite
number of degrees of freedom...
...We have shown how an approach to quantum mechanics can be built from the algebraic
structure of the Clifford algebra and the discrete Weyl algebra (or the generalised
Clifford algebra). These algebras can be treated by the same techniques that do not
require Hilbert space yet enable us to calculating mean values required in quantum
mechanics...


------------
Algebra approach to Quantum Mechanics A: The Schroedinger and Pauli Particles.
http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4031v1.pdf


.

 

[unusualname]

Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space.
http://www.bbk.ac.uk/tpru/BasilHiley/Algebraic Quantum Mechanic 5.pdf

...These results have only intensified my curiosity as to why most if not all of the results
can be obtained without seemingly the need to resort to Hilbert space. This goes against
the prevailing orthodoxy that appears to insist that quantum mechanics cannot be done
except in the context of a Hilbert space. Yet there have been other voices raised against
the necessity of Hilbert space. Von Neumann himself wrote to Birkoff (1966) writing "I
would like to make a confession which may seem immoral: I do not believe absolutely in
Hilbert space any more." (A detailed discussion of why von Neumann made this
comment can be found in Rédei 1996).
But there are more important reasons why an algebraic approach has advantages. As
Dirac (1965) has stressed, when algebraic methods are used for systems with an infinite
number of degrees of freedom...
...We have shown how an approach to quantum mechanics can be built from the algebraic
structure of the Clifford algebra and the discrete Weyl algebra (or the generalised
Clifford algebra). These algebras can be treated by the same techniques that do not
require Hilbert space yet enable us to calculating mean values required in quantum
mechanics...------------
Algebra approach to Quantum Mechanics A: The Schroedinger and Pauli Particles.
http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4031v1.pdf.

yeah, I think they were worried about the "completeness" part of Hlibert Space not being a fundamental requirement (to describe reality), they'll probably be proved right soon.

 

[Chopin]

I guess I'm curious why folks seem to be so set on finding a way to do QM without a Hilbert space in the first place. Is there something distasteful about using it? Even if it is possible to do, whatever other mechanism you come up with is going to have to reproduce many of the features of a Hilbert space, so it's not like you can get around understanding how it works. The reason we use it is that the main feature of quantum mechanics--superposition--is handled very naturally by the linear algebra of a Hilbert space.

So if your motivation in this is that you saw some examples that dealt with state vectors, linear operators, or bra-ket notation and decided that you didn't want to learn all of that stuff, then you're going to be disappointed. There aren't any shortcuts on this one--you've got to learn how all that works if you want to have any kind of grasp on quantum mechanics. Understanding why a Hilbert space is so useful for describing the features of QM is a necessary step before you're going to have any hope of finding another way of doing it.

 

[Varon]

I guess I'm curious why folks seem to be so set on finding a way to do QM without a Hilbert space in the first place. Is there something distasteful about using it? Even if it is possible to do, whatever other mechanism you come up with is going to have to reproduce many of the features of a Hilbert space, so it's not like you can get around understanding how it works. The reason we use it is that the main feature of quantum mechanics--superposition--is handled very naturally by the linear algebra of a Hilbert space.

So if your motivation in this is that you saw some examples that dealt with state vectors, linear operators, or bra-ket notation and decided that you didn't want to learn all of that stuff, then you're going to be disappointed. There aren't any shortcuts on this one--you've got to learn how all that works if you want to have any kind of grasp on quantum mechanics. Understanding why a Hilbert space is so useful for describing the features of QM is a necessary step before you're going to have any hope of finding another way of doing it.

What if one learns QM by starting with Hilbert Space and then latter the plain Fourier based Wave function like depicted in the site:
http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html

I'm more familiar with Hilbert space having started understanding QM comtemplating on the double slit experiment and Schrodinger Cat in superposition.. so I know about pure state, mixed state, decoherence.

I won't go into full details of QM as I'm not a physics student. I just need a basic mathematical grasp to understand the different interpretations as the reason we novices get into QM is because of the weirdness in it.. lol... so Hilbert space is sufficient for us.. right?

 

[Varon]

I mean I just want to use vector space. Most course in QM started with the particle in a box and hamiltonian and energy and all that and it takes a long time. Since I just want to focus on Schroedinger cat.. I don't need to use energy.. but just position.. so I wonder if I can just focus on Hilbert space or vector space and later go to Hamiltonian of energy of electrons in atoms.. which I don't really need or not urgent right now..

 

[Varon]

But I have read the best novice book on particle physics. Deep Down Things. It summarizes what is the Schroedinger Equation:

"Thus, the Schroedinger Equation is just the wave-mechanical statement that the sum of the kinetic and potential energies at any given point is just equal to the total energy - the Schroedinger equation is simply the quantum-mechanical version of the notion of energy conservation."

Is the "energy" bias is because particle physics is the application? I can't find the thread where I read this (if anyone know the thread title where this argument is given that the "energy' is due to the some kind of bias, pls. let me know). The book never mentions about Hilbert Space or about the cat or double slit. It goes on to talking about Quantum Field theory and Gauge theory. So it seems the Schroedinger equation has two major usages. In particle physics to determine the kinetic energy, potential, etc. of particles. and in general case, to determine superpositions like in double slit. But maybe one can give a general statement of what is the Schrodinger equation aside from merely telling it is "simply the quantum-mechanical version of the notion of energy conservation"

 

[Fredrik]

The most explicit connection between the Schrödinger equation and energy conservation is that in wave mechanics (the quantum theory of a single spin-0 particle in Galilean spacetime), "momentum" and "energy" are represented by the operators

-i\nabla,\quad i\frac{d}{dt}

If you just take the classical relation E=p²/2m, and make the substitutions

p\rightarrow-i\nabla,\quad E\rightarrow i\frac{\partial}{\partial t}

what you get is the Schrödinger equation (without a potential). (I believe that this is what your book is referring to). I don't like to emphasize this too much, because this substitution trick doesn't work in special relativistic QM, not even for single particle theories. (You get a useful equation, but the function it acts on can't be interpreted as a wavefunction).

A more sophisticated approach (which works with both non-relativistic and special relativistic QM) is to start by noting that there must exist a time evolution operation U(t) (for each t) that takes a state vector to the state vector that represents the state of the system time t later, and it must satisfy U(t+s)=U(t)U(s). It's possible to show that this condition implies that there exists a self-adjoint operator H such that U(t)=e^{-iHt}. This H is called the Hamiltonian. (This should be taken as the definition of H). The above implies U'(t)=-iHU(t), or equivalently,

i\frac{dU(t)}{dt}=HU(t)

This is another version of the Schrödinger equation. If you define \psi(x,t)=U(t)\psi(x,0) (this is only done in non-relativistic QM), and have the operators in the equation above act on \psi(x,0), you get the more familiar version

i\frac{\partial\psi(x,t)}{\partial t}=H\psi(x,t)

You asked for a general statement of what the Schrödinger equation is. I would just say that it's the equation that tells us how state vectors change with time. So in addition to being the QM counterpart of E=p²/2m, it's also the QM counterpart of F=ma. (Newton's 2nd is what tells us how states change with time in classical mechanics).

 

[Varon]

The most explicit connection between the Schrödinger equation and energy conservation is that in wave mechanics (the quantum theory of a single spin-0 particle in Galilean spacetime), "momentum" and "energy" are represented by the operators

-i\nabla,\quad i\frac{d}{dt}

If you just take the classical relation E=p²/2m, and make the substitutions

p\rightarrow-i\nabla,\quad E\rightarrow i\frac{\partial}{\partial t}

what you get is the Schrödinger equation (without a potential). (I believe that this is what your book is referring to). I don't like to emphasize this too much, because this substitution trick doesn't work in special relativistic QM, not even for single particle theories. (You get a useful equation, but the function it acts on can't be interpreted as a wavefunction).

A more sophisticated approach (which works with both non-relativistic and special relativistic QM) is to start by noting that there must exist a time evolution operation U(t) (for each t) that takes a state vector to the state vector that represents the state of the system time t later, and it must satisfy U(t+s)=U(t)U(s). It's possible to show that this condition implies that there exists a self-adjoint operator H such that U(t)=e^{-iHt}. This H is called the Hamiltonian. (This should be taken as the definition of H). The above implies U'(t)=-iHU(t), or equivalently,

i\frac{dU(t)}{dt}=HU(t)

This is another version of the Schrödinger equation. If you define \psi(x,t)=U(t)\psi(x,0) (this is only done in non-relativistic QM), and have the operators in the equation above act on \psi(x,0), you get the more familiar version

i\frac{\partial\psi(x,t)}{\partial t}=H\psi(x,t)

You asked for a general statement of what the Schrödinger equation is. I would just say that it's the equation that tells us how state vectors change with time. So in addition to being the QM counterpart of E=p²/2m, it's also the QM counterpart of F=ma. (Newton's 2nd is what tells us how states change with time in classical mechanics).

You mentioned "state vector". So your formulation is in Hilbert Space format? But almost all introductory QM doesn't use any Hilbert Space. See:

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html

Without state vector. What is the equivalent of your formulation in the same mathematical language as the web site? Many thanks.

 

[Chopin]

You mentioned "state vector". So your formulation is in Hilbert Space format? But almost all introductory QM doesn't use any Hilbert Space. See:

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html

Without state vector. What is the equivalent of your formulation in the same mathematical language as the web site? Many thanks.

\Psi is a state vector. It's a state in a Hilbert space, they're just going to lengths to avoid calling it that. When we specify the state of a system with a wavefunction, what we're saying is that the system is in a superposition of position states--specifically, that the probability that the particle is at x_0 is |\Psi(x_0)|^2, the probability that it's at x_1 is |\Psi(x_1)|^2, etc.

We can cast this into bra-ket notation (the language of the Hilbert space) by denoting |x\rangle as the state with position x. If we do so, then our particle's state is simply \Psi(x_0)|x_0\rangle + \Psi(x_1)|x_1\rangle + .... Or, put more simply, \int{\Psi(x) |x\rangle dx}. So in this case, wavefunctions and Hilbert spaces are exactly equivalent.

If you're looking at a text that doesn't talk about Hilbert spaces, they're just hiding the concept inside other terms. For instance, the site you linked to talks about building up a wavefunction out of multiple basis states, called \Psi_1, \Psi_2, \Psi_3, .... These are exactly the same thing as state vectors--you could just as easily call them |a_1\rangle, |a_2\rangle, |a_3\rangle, .... The math is the same.

Hilbert spaces are just more general than the wavefunction--they allow you to talk about discrete-valued operators like spin and polarization, they let you talk about situations with multiple species of particle, and they let you talk about situations where the number of particles changes (for instance, pair production.) The simple wavefunction can do none of these things. If you move on to advanced quantum mechanics or quantum field theory, you will find no mention of the wavefunction anymore--it's all bras and kets. For instance, in quantum field theory, the equation for determining how an initial state transforms into a final state is given by:

\langle\phi_{final}|T(e^{i\int{H(t) dt}})|\phi_{initial}\rangle

There's no \Psi--only bras and kets. So if you really want to understand quantum mechanics at anything past a basic level, you're going to have to learn how to do this stuff. The sites you mentioned are doing basic enough stuff that they don't need to get into it, but if you understand all of that, then learning this should be your next step.

 

[Fredrik]

I wouldn't say that the introductory texts aren't using Hilbert spaces. They kind of are. What they're doing is very close to introducing a specific Hilbert space, instead of stating the general definition of a Hilbert space and leaving the details of what Hilbert space we're dealing with unspecified. I'm saying "very close to", because they're leaving out one important detail, which will be mentioned below.

They're considering functions from \mathbb R^3 into \mathbb C that are square-integrable, i.e. if f is such a function, then

\int |f(x)|^2 d^3x

is a real number. The set of such functions is given the structure of a vector space by the "obvious" definitions of addition and scalar multiplication:

(f+g)(x)=f(x)+g(x)

(af)(x)=a(f(x))

So these functions are vectors, in the sense "members of a vector space". We can define a semi-inner product and a semi-norm by

\langle f,g\rangle =\int f(x)^*g(x) d^3x

\|f\|^2=\langle f,f\rangle

The former fails to satisfy the definition of an inner product, and the latter fails to satisfy the definition of a norm, because this condition isn't satisfied:

\langle f,f\rangle=0\ \Rightarrow f=0

For example, if we define f(x)=1 when x=(x_1,x_2,x_3) is a triple of rational numbers, and f(x)=0 otherwise, then \|f\|=0.

The detail I had in mind when I said that there's one thing the introductory books are leaving out is the trick that gives us a Hilbert space:

We define an equivalence relation on this set of functions, by saying that f and g are equivalent if and only if the set where they take different values has Lebesgue measure 0. (Roughly speaking, this means that it has a well-defined size, and that that size is 0). In particular, since the set of numbers with rational coordinates has Lebesgue measure 0, the function that has the value 1 on that set and 0 everywhere else, belongs to the same equivalence class as the constant function that has the value 0 everywhere.

The set of such equivalence classes is denoted by L^2(\mathbb R^3). Let's denote the equivalence class that f belongs to by [f], and define

a[f]+b[g]=[af+bg]

\langle [f],[g]\rangle=\langle f,g\rangle

\|[f]\|=\|f\|

Now we have a space with an actual inner product, and an actual norm, and it happens to be one of the simplest examples of an infinite-dimensional Hilbert space.

So the space that the introductory books are working with is a semi-inner product space that can be turned into a Hilbert space quite easily. The reason why they can work with this semi-inner product space instead of the Hilbert space is a) that many results that hold for Hilbert spaces hold for semi-inner product spaces too (e.g. the Cauchy-Bunyakovsky-Schwarz inequality), and b) that they're not going to do things rigorously anyway. These things make the differences between these two spaces pretty much irrelevant.

 

[Varon]

Thanks for the valuable input that many introductory quantum text without Hilbert space doesn't even explain the connection.

Anyway. There is an issue of primary importance. Many said that the state vector and collapse is just our knowledge of the system. But in the double slit, the detector indeed collapse the wave function even if you don't do calculations. So collapse exists independently from your knowledge. What do you say about this?