(This was posted in the welcome thread, and I received a PM suggesting I post it here.)
I am looking for someone to help me with the Quantum Physics sequence. I have little background in physics and mathematics. For purposes of the sequence, you could probably consider me “intelligent but uninformed” or something like that.
To indicate the level on which I am having difficulties, take as an example the Configurations and Amplitude post.
I can do the algebra involved.
I found the articles linked in this comment helpful.
I understand the notion of configuration space in its classical sense.
I do not understand how the term “amplitude” is functioning in the post.
Hopefully that gives you an idea of where I am at and what sort of help I might need. It’s totally basic stuff, obviously, and there is a part of me that is somewhat embarrassed to ask. Nevertheless, learning is more important than avoiding embarrassment.
What sort of help are you looking for? Do you have specific questions (like, do you want someone to explain the notion of amplitude) or are you looking for general resources on QM?
If the latter, then I highly recommend Leonard Susskind’s lectures on quantum mechanics from his “Theoretical Minimum” series, available here. Susskind does assume that you know calculus. If you don’t, then I suggest that you familiarize yourself with calculus before attempting a technical understanding of QM.
If you’d rather read than watch, the lectures are also available in book form, here.
I do not know calculus, but I am convinced that I need to for a variety of reasons, so I have begun working my way through the Khan Academy materials. I had intended to leave the quantum physics materials aside until that project was complete, but I was heartened by Eliezer’s insistence that one need only know algebra to grasp the sequence. Perhaps I just need to do calculus first, then work through a few books/lectures. Do you think this to be the case?
I don’t recommend Eliezer’s sequence as a first introduction to QM, at least not if you’re interested in developing a reasonably deep understanding of the theory. If you want a minimal-math introduction, I think a better bet would be to check out the first few chapters of David Albert’s Quantum Mechanics and Experience, illegally available in it’s entirety here. I don’t think Albert’s book is an ideal introduction either, but I do think it does a better job than the sequence at getting the salient points across in simple language. Also, since you’re interested in ontology, the latter half of Albert’s book contains a pretty incisive analysis of various interpretations of QM (not just MWI).
And here’s an attempt at explaining quantum amplitudes:
Let’s start with classical configuration space, since you understand that. The possible states of a classical system are represented by individual points in its configuration space (well, technically, the configuration space doesn’t give you the complete state of the system, because it leaves out information about velocities, but let’s ignore that for now).
In quantum mechanics, the configuration space looks just like classical configuration space, but its interpretation is very different. It’s no longer true that the state of a quantum system is represented by an individual point in configuration space. The state of a quantum system is represented by a function on configuration space (the function has to satisfy certain other requirements in order to qualify as a bona fide representation of a quantum state, but ignore that detail also).
So imagine your configuration space is only two-dimensional for now, and again imagine that we are only considering real-valued functions on configuration space. In that case, you could construct 3-D plots of the various functions that correspond to quantum states, with the x-y plane representing the configuration space and the z axis representing the value of the function. Here’s an example of one such function (this function actually doesn’t satisfy those extra requirements I mentioned above, but we’re ignoring those). Looks kind of like a wave, doesn’t it?
So that function represents one possible state of the quantum system. Other possible states are represented by other functions. A quantum state is simply a function mapping each point on configuration space to a number; in other words, it associates a single number with every individual classical configuration. This number is what we call the amplitude of that particular configuration. Go back to our 3-D plot from above. The amplitude of any particular configuration (any point on configuration space) is just the height of our wave-like function at that point. And the quantum state is just the collection of all these amplitudes.
Now things are a little more complicated in quantum mechanics because the functions we consider aren’t necessarily real-valued. They don’t have to associate a real number with each point on configuration space. They are actually complex-valued functions, which means that each point in configuration space is associated with a complex number. That’s no longer easy to plot in three dimensions unfortunately, so you’ll have to use your imagination a little bit. And of course, the configuration spaces for actual systems we are interested have many many more dimensions than just two.
Now what the amplitude does is give you the probability of observing some particular configuration when you decide to look at the system.In order to figure out the probability of seeing a certain configuration, take the complex number associated with that configuration by the quantum state, and then calculate its squared magnitude. That will give you the probability.
That is extremely helpful; it is just the kind of explanation I was looking for. I have begun working through some of the materials linked here, as well. Many thanks. Now that I am starting to piece the picture together, I need some time to mull over it and let my intuitions adjust to the ideas, but I may send you a message when I next get hung up on it.
If you have specific questions to ask, I’d be happy to answer them.
“Amplitude” just refers to a complex number corresponding to a given point in configuration space. The Schrödinger equation specifies how the field of these complex numbers evolves over time. The probability of being in a particular configuration is proportional to the square of this amplitude. Does that help?
I recommend that you first read popular or semi-popular books written by experts in the field (Eliezer isn’t one). One of the more recent and highly praised semi-popular books which addresses many points Eliezer tried to get across is ScottAaronson’s Quantum Computing since Democritus. Free lecture notes are also available, but not as complete. The book has a complexity-theoretic bend, but you can skip the parts you find too boring or too hard. Other classic semi-popular QM books are also available, including the venerable Feynman lectures. That one explains amplitudes very well, but is light on various ontologies, like MWI.
While Aaronson’s book is excellent, I suspect that someone who had trouble following Eliezer’s posts will also have trouble following Aaronson’s discussion of QM. It’s not very neophyte-friendly.
Yeah, it requires effort, but, unlike Susskind’s book, it has basically no calculus, just algebra and a tiny bit of some basic matrix addition and multiplication, as well as some very brief understanding of complex numbers, both of which can be learned in an afternoon by a person who has a solid grasp of precalc (grade 11-level math or so). Basically the same prereqs as for the QM sequence. Additionally, it touches on several very AGI-relevant points, like Godel incompleteness, anthropics, complexity and free will. And, while it talks favorably about MWI, it has none of the anti-rational MWI/Bayes propaganda and “eld science” bashing of the QM sequence.
And, while it talks favorably about MWI, it has none of the anti-rational MWI/Bayes propaganda and “eld science” bashing of the QM sequence.
Of course, it throws in some gratuitous anti-Bayesianism too—remember the chapter where anthropics (which no one agrees on or can formulate a sensible position on) refutes Bayesianism? Pick your poison...
It is the ontology angle in which I am most interested, but I am not convinced that I can understand the ontology on even a basic level without understanding the math.
(This was posted in the welcome thread, and I received a PM suggesting I post it here.)
I am looking for someone to help me with the Quantum Physics sequence. I have little background in physics and mathematics. For purposes of the sequence, you could probably consider me “intelligent but uninformed” or something like that.
To indicate the level on which I am having difficulties, take as an example the Configurations and Amplitude post.
I can do the algebra involved.
I found the articles linked in this comment helpful.
I understand the notion of configuration space in its classical sense.
I do not understand how the term “amplitude” is functioning in the post.
Hopefully that gives you an idea of where I am at and what sort of help I might need. It’s totally basic stuff, obviously, and there is a part of me that is somewhat embarrassed to ask. Nevertheless, learning is more important than avoiding embarrassment.
What sort of help are you looking for? Do you have specific questions (like, do you want someone to explain the notion of amplitude) or are you looking for general resources on QM?
If the latter, then I highly recommend Leonard Susskind’s lectures on quantum mechanics from his “Theoretical Minimum” series, available here. Susskind does assume that you know calculus. If you don’t, then I suggest that you familiarize yourself with calculus before attempting a technical understanding of QM.
If you’d rather read than watch, the lectures are also available in book form, here.
I was looking for something more like the former.
I do not know calculus, but I am convinced that I need to for a variety of reasons, so I have begun working my way through the Khan Academy materials. I had intended to leave the quantum physics materials aside until that project was complete, but I was heartened by Eliezer’s insistence that one need only know algebra to grasp the sequence. Perhaps I just need to do calculus first, then work through a few books/lectures. Do you think this to be the case?
I don’t recommend Eliezer’s sequence as a first introduction to QM, at least not if you’re interested in developing a reasonably deep understanding of the theory. If you want a minimal-math introduction, I think a better bet would be to check out the first few chapters of David Albert’s Quantum Mechanics and Experience, illegally available in it’s entirety here. I don’t think Albert’s book is an ideal introduction either, but I do think it does a better job than the sequence at getting the salient points across in simple language. Also, since you’re interested in ontology, the latter half of Albert’s book contains a pretty incisive analysis of various interpretations of QM (not just MWI).
And here’s an attempt at explaining quantum amplitudes:
Let’s start with classical configuration space, since you understand that. The possible states of a classical system are represented by individual points in its configuration space (well, technically, the configuration space doesn’t give you the complete state of the system, because it leaves out information about velocities, but let’s ignore that for now).
In quantum mechanics, the configuration space looks just like classical configuration space, but its interpretation is very different. It’s no longer true that the state of a quantum system is represented by an individual point in configuration space. The state of a quantum system is represented by a function on configuration space (the function has to satisfy certain other requirements in order to qualify as a bona fide representation of a quantum state, but ignore that detail also).
So imagine your configuration space is only two-dimensional for now, and again imagine that we are only considering real-valued functions on configuration space. In that case, you could construct 3-D plots of the various functions that correspond to quantum states, with the x-y plane representing the configuration space and the z axis representing the value of the function. Here’s an example of one such function (this function actually doesn’t satisfy those extra requirements I mentioned above, but we’re ignoring those). Looks kind of like a wave, doesn’t it?
So that function represents one possible state of the quantum system. Other possible states are represented by other functions. A quantum state is simply a function mapping each point on configuration space to a number; in other words, it associates a single number with every individual classical configuration. This number is what we call the amplitude of that particular configuration. Go back to our 3-D plot from above. The amplitude of any particular configuration (any point on configuration space) is just the height of our wave-like function at that point. And the quantum state is just the collection of all these amplitudes.
Now things are a little more complicated in quantum mechanics because the functions we consider aren’t necessarily real-valued. They don’t have to associate a real number with each point on configuration space. They are actually complex-valued functions, which means that each point in configuration space is associated with a complex number. That’s no longer easy to plot in three dimensions unfortunately, so you’ll have to use your imagination a little bit. And of course, the configuration spaces for actual systems we are interested have many many more dimensions than just two.
Now what the amplitude does is give you the probability of observing some particular configuration when you decide to look at the system.In order to figure out the probability of seeing a certain configuration, take the complex number associated with that configuration by the quantum state, and then calculate its squared magnitude. That will give you the probability.
Hope that helped a little.
(Sorry for the delay in response.)
That is extremely helpful; it is just the kind of explanation I was looking for. I have begun working through some of the materials linked here, as well. Many thanks. Now that I am starting to piece the picture together, I need some time to mull over it and let my intuitions adjust to the ideas, but I may send you a message when I next get hung up on it.
If you have specific questions to ask, I’d be happy to answer them.
“Amplitude” just refers to a complex number corresponding to a given point in configuration space. The Schrödinger equation specifies how the field of these complex numbers evolves over time. The probability of being in a particular configuration is proportional to the square of this amplitude. Does that help?
I recommend that you first read popular or semi-popular books written by experts in the field (Eliezer isn’t one). One of the more recent and highly praised semi-popular books which addresses many points Eliezer tried to get across is ScottAaronson’s Quantum Computing since Democritus. Free lecture notes are also available, but not as complete. The book has a complexity-theoretic bend, but you can skip the parts you find too boring or too hard. Other classic semi-popular QM books are also available, including the venerable Feynman lectures. That one explains amplitudes very well, but is light on various ontologies, like MWI.
While Aaronson’s book is excellent, I suspect that someone who had trouble following Eliezer’s posts will also have trouble following Aaronson’s discussion of QM. It’s not very neophyte-friendly.
Yeah, it requires effort, but, unlike Susskind’s book, it has basically no calculus, just algebra and a tiny bit of some basic matrix addition and multiplication, as well as some very brief understanding of complex numbers, both of which can be learned in an afternoon by a person who has a solid grasp of precalc (grade 11-level math or so). Basically the same prereqs as for the QM sequence. Additionally, it touches on several very AGI-relevant points, like Godel incompleteness, anthropics, complexity and free will. And, while it talks favorably about MWI, it has none of the anti-rational MWI/Bayes propaganda and “eld science” bashing of the QM sequence.
Of course, it throws in some gratuitous anti-Bayesianism too—remember the chapter where anthropics (which no one agrees on or can formulate a sensible position on) refutes Bayesianism? Pick your poison...
I don’t think he ever said anything about “refuting” Bayesianism, only that its application may depend on whether you believe SIA or SSA.
It is the ontology angle in which I am most interested, but I am not convinced that I can understand the ontology on even a basic level without understanding the math.