Difference between measurement, state and system in quantum mechanics

Question

This question refers to the following point made in Susskind's book Quantum Mechanics- The Theoretical Minimum:

In the classical world, the relationship between the state of a system and the result of a measurement on that system is very straightforward. In fact, it’s trivial. The labels that describe a state (the position and momentum of a particle, for example) are the same labels that characterize measurements of that state. To put it another way, one can perform an experiment to determine the state of a system. In the quantum world, this is not true. States and measurements are two different things, and the relationship between them is subtle and nonintuitive.

I'm not sure I understand the last line since it seems to imply that states and measurements are not "different things" in the classical realm. Are both of them the "same" in the sense that they both refer to a point in the system's phase space?

Of course, any particular state would uniquely specify a measurement in classical mechanics and conversely a set of measurements would uniquely specify a state. Such a correspondence doesn't exist in quantum mechanics. So is that what the author means by "states and measurements are two different things"?

Finally, what does the author mean by "labels"? Do they simply refer to the values of the various degrees of freedom of the system?

Now, coming to the next part:

Attached to the electron is an extra degree of freedom called its spin. [...] We can and will abstract the idea of a spin, and forget that it is attached to an electron. The quantum spin is a system that can be studied in its own right.

Why is the spin being called a "system"? Isn't a system supposed to be something physical instead of a mathematical abstraction? (And it's defined as a degree of freedom in the first place- from what I understand, a degree of freedom is meant to characterize a physical system.)

Answer 1

Are both of them the "same" in the sense that they both refer to a point in the system's phase space?

Yes, this is how we identify a point in the system's phase space. We look for what measurements uniquely define a state, and the results of these measurements are used to describe that state. This only works because if the the system is at a specific point in phase space, it will - classically - always yield the same measurement. This is related to:

Finally, what does the author mean by "labels"? Do they simply refer to the values of the various degrees of freedom of the system?

Yes. For a free particle, these labels might be "position" (times 3), and "momentum" (times 3). Since these labels uniquely define the state, and we can also perform measurements on these quantities that - given the same state - will always yield the same result, we can use the results of the measurements to uniquely define a point in phase space. That is to label them with the measurement results.

Such a correspondence doesn't exist in quantum mechanics. So is that what the author means by "states and measurements are two different things"?

Exactly. Because things are no longer deterministic (for non-commuting operators, anyway), we can no longer use the results of measurements to define a state, because even if two systems are in the same quantum state (ignoring the Pauli principle for a second), if we measure all degrees of freedom for both systems, the results will almost certainly vary. So the results of all possible measurements are no longer suitable to identify a state.

Some measurements are still usable as identifiers ("labels"), if the system is in a certain state - a so-called eigenstate of this operator. These are then called quantum numbers. Complications arise because some measurements influence each other - that is when they do not commute. Whenever this is the case, e.g. position and momentum, the system cannot be in an eigenstate of both of these operators. This is why we can no longer use all possible measurements as identifiers. Furthermore, a general state is a linear combination of eigenstates of some set of operators, which also leads to varying outcomes of measurements.

Why is the spin being called a "system"?

This is a bit of hair-splitting the terminology. Usually, a system is some physical part of the universe that we want to investigate and describe, and it interacts with (or is maybe isolated from) the environment. In this specific case it means that we can decouple the description of some physical thing (the spin) from a different physical thing (the electron) it is usually attached to.

Answer 2

Let us start from what a label is. In classical physics, the label of a given state is a possible value of some dynamical variables and therefore it is an outcome of a measurement. For example the direction $\hat n$ of the magnetization vector of a piece of iron is a label. We can easily find this direction by means of measurements which do not change it.

In quantum mechanics a label of a state is also given by the outcomes of a measurement. However not every measurement can be used to label a state. Only those resulting in good quantum numbers are labels and this implies that in general it is not possible to use any measurement to find out the state. The reason is that by making a measurement in a quantum system we in general change it. For example if we prepare the spin of some system to be in the $+z$ direction we know for sure that its state is $|+z\rangle$. If we orient the measurement apparatus to measure spin $z$ component, $S_z$, we definitely find $+1$ which means we succeed in finding the state by means of a measurement. The possible outcomes of $S_z$ can be used to label the state. On the other hand a measurement of $S_x$ cannot be used as a label of that already $+z$ prepared system since it will interfere drastically with the system. Classically you would expect that since there is no spin along $x$ then a measurement along $x$ would give $0$. However the measurement $S_x$ gives either $+1$ or $-1$ which means the system was projected into a new state, $|+x\rangle$ or $|-x\rangle$.

Answer 3

It's a big difference.

In classical mechanics the entities that define the state are also the same ones that one measures. Not only that, one can simultaneously measure multiple parameters. Examples are position x and momentum p, which in classical physics are measured without one affecting the other. Not only that, the state in classical mechanics are indeed defined by the values of x and p. Newton's equations guarantee that if we know the state x and p at any one time, and know the force acting on it at that time, we can obtain x and p (the state) at an infinitesimally small time later. Thus, we can predict the trajectory if we always know the force. And of course we can measure the parameters x and p at all times later.

The labels he refers to are the parameters that define the state. As he says, it's x and p.

In quantum mechanics (QM) the state is not x and p, but the wavefunction $\psi$. It can also be written as a key vector, the Dirac ket vector. In QM the wavefunction, or a ket vector, can be written in terms of a basis. The simplest equivalent to the classical example is a position basis (but we could do it using the ket notation and leave it as an abstract vector), and in the position basis the wavefunction is a function of position, x, for ALL values of x. That is the state in the position basis. It can also be written in the momentum basis, and it turns out to be related to the position wavefunction by Fourier transforms. They both represent the same state. So, a wavefunction is basically what is called an element of a Hilbert space. The labels for the state in QM are vectors in a Hilbert space.

But the measurements are still an attempt to obtain some values for x and p. The wavefunction gives us probabilities that x is in a certain range of x's, or that p is in some range. Further, we cannot measure both at the same time becuase they affect each other; in QM parlance, x and p are operators defining possible observables, but they do not commute with each other. So they each interfere with the measurement of the other (btw, in three dimensions it is position and momentum in each direction which do not commute and are related). So what do we measure: when we try to measure x we do get a result, following the probabilities, where we observe an eigenvalue (or a quantum number) of the operator x. If we repeated the same experiment prepared exactly the same way to start with, we'd measure different values of x, with probabilities determined by the wavefunction.

So, yes, that's why it is very different than in classical physics. You measure and obtain an eigenvalue, and if you can repeat the experiment many times, you'd get the probability function derived from the wave function (it's absolute value squared).

That is why Susskind said they are very different things, and very counterintuitive.

What are the labels that describe the state in QM? The wavefunction or ket vectors, generally can be said to be a vector in a Hilbert space. Now you need to actually go learn all that. I'm sure Susskind covers that in the book and his lectures, jsust persevere. He also tries hard to give you hints of what's coming and later explains it very well. And you can always read some other QM book

Spins are like x or p, except they have a discrete number of eigenvalues, or possible states. Usually it's 1/2 and -1/2, for spin 1/2 particles, but spin can be any half integer. Spin can be analyzed in simple cases independent of the other degrees of freedom, so he just uses the system word. Easier that way,mbecause spin is even more counterintuitive. To learn anything just follow the whole course