What is Quantum states: Definition and 84 Discussions
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vectors are identified by the principal quantum number n, the angular momentum quantum number l, the magnetic quantum number m, and the spin zcomponent sz. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. The Hilbert space for the electron's spin is therefore twodimensional, constituting a qubit. A pure state here is represented by a twodimensional complex vector
(
α
,
β
)
{\displaystyle (\alpha ,\beta )}
, with a length of one; that is, with

α

2
+

β

2
=
1
,
{\displaystyle \alpha ^{2}+\beta ^{2}=1,}
where

α

{\displaystyle \alpha }
and

β

{\displaystyle \beta }
are the absolute values of
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
. A mixed state, in this case, has the structure of a
2
×
2
{\displaystyle 2\times 2}
matrix that is Hermitian and positive semidefinite, and has trace 1. A more complicated case is given (in bra–ket notation) by the singlet state, which exemplifies quantum entanglement:

ψ
⟩
=
1
2
(

↑↓
⟩
−

↓↑
⟩
)
,
{\displaystyle \left\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\big (}\left\uparrow \downarrow \right\rangle \left\downarrow \uparrow \right\rangle {\big )},}
which involves superposition of joint spin states for two particles with spin 1⁄2. The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability.
A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. The Schrödinger–HJW theorem classifies the multitude of ways to write a given mixed state as a convex combination of pure states. Before a particular measurement is performed on a quantum system, the theory gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the linear operators describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the uncertainty principle: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another.
Ballentine, in his Chapter 8.1, appears to give the attached recipe for *in principle* preparing an (almost) arbitrary (pure) state (of a particle with no internal degrees of freedom) by the method of "waiting for decay to the energy ground state". My questions are fourfold:
1) From (8.1), we...
This pop article popped up (isn't that what they do, by definition?) on my google news page.
https://www.sciencenews.org/article/blackholeparadoxesquantumstates
It claims that a thought experiment shows that doing a doubleslit experiment near a black hole event horizon can reveal...
What's the difference between a bra vector and ket vector in specifying spin states except for notational convenience when calculating probablility amplitudes? Are they equivalent?
I have some basic questions about mixed states and entanglement.
1. Do mixed states always imply that the states are entangled and vice versa?
2. Can mixed states ever be separable?
3. Does interference have anything to do with entanglement?
In terms of Density Matrices, ρ = ψ><ψ:
4...
I was working on plotting fidelity with time for two quantum states. First I used discrete time( t= 0,1,2,3...etc) to plot my fidelity. I got constant fidelity as 1 with continuous value of time. Next I used discrete set of values ( t=0 °,30 °,60 °,90 °). Here I saw my fidelity decreases and...
I was teaching the basics of quantum states and was showing the students that an arbitrary state in a quantum twolevel system could be written as ##\psi\rangle = C_1 +\rangle + C_2 \rangle = R_1 +\rangle + R_2 e^{i \alpha} \rangle##, with {##C_i##} complex and {##R_i##} real.
Then...
hi guys
i have a question about the derivation of the density of states , after solving the Schrodinger equation in the 3d potential box and using the boundary conditions ... etc we came to the conclusion that the quantum state occupy a volume of ##\frac{\pi^{3}}{V_{T}}## in k space
and to...
I'm trying to understand the detailed concept of why the density of states formula is accurate enough to calculate the number of quantum states of an energy level, including degeneracy, within a small energy interval of ##dE##.
The discrete energie levels are calculated by
$$E = \frac{h^2 \cdot...
Hi all,
This question asks me to calculate the number of quantum states, as well as electrons per cm^3 of the crystal in the room temperature.
The problem is I only dealt with a single element before without any calculation for 1cm^3 whatsoever. For example for a Silicon semiconductor, I can...
Hi all,
I'm right now confused about this.
As far as I know, when changing from a level to another, the change in l (subshell) can only be a difference of 1, and ##m_{l}## can be the same or a difference of 1.
In this case, since the question wants me to state possible quantum states of...
I calculated the total area of phase space and divided it by the area of one cell i.e. h.
n = (x_0*m*2*v)/h
=> n = (0.1 x 10^10 x 9.1 x 10^31 x 2 x 10^7)/6.626 x 10^34
=> n = 0.27
This answer doesn't match with any of the options. What did I do wrong?
Edit: The question was printed...
Elementary question: Is there ever a case where the solutions for a wave equation turn out not to be a vector (in Hilbert space of infinite complexvalued dimensions, or a restriction to a subspace thereof) , but something else  say, (higherorder) tensors or bivectors, or some such?
My...
Summary: Finding state at t=0, energy values and more
So this is my first question in quantum mechanics (please understand).
1. So we have a system, and to describe the state of the system we have to measure, A is an hermitian matrix, that each physical measurable quantity has.
To find the...
Hello,
I have some trouble understanding how to construct the matrix for the beam splitter (in a MachZehnder interferometer).
I started with deciding my input and output states for the photon.
I then use Borns rule, which I have attached below:
To get the following for the state space...
Hi!
I'm struggling with the following question:
Show that if n quantum states ρ1, ..., ρn are pairwise perfectly distinguishable, they are also jointly perfectly distinguishable.
Perfect distinguishability means that there is a set of psd matrices \{E_{1}, ..., E_{n}\},\, \sum_{i} E_{i} =...
In my textbook, quantum states are infinite dimensional vectors. But I was watching a lecture on QM and the professor referred to ##v> <u## as itself being a quantum state. Also I saw online people saying the same thing.
Are tensor products just things that tell you whether or not the two...
EDIT: Questions have been revised below, those immediately following are for reference, jambaugh's kind reply was in direct response to these original questions.
Could a completely unitary (QM) process act on a set of particles in "completely identical quantum states" to cause them to time...
Hello everyone,
We come to the end of another semester and its presentation time. I have chosen to discuss how to prepare different quantum mechanical systems for various applications.
So my question for you guys is, are there any interesting experimental techniques I should look into. I am...
If the number of possible values of L is n, and the number of possible values of m is 2*L1, and there are 2 spin directions.. shouldn't the total number of states be 2*(number of possible L)*(Number of possible m)? But this gives 4n^2  2n. I am extremely confused. Thanks for your help!
Hi,
I am learning quantum entanglement. I am interested to create an up to date list of all known :
 Photon Quantum States
 Particle Quantum States
 Classically entagled photon states
I guess that there is an organization out there that already have this info.
If someone can point me into...
Hello everyone,
My understanding is that a twoquantum state system is simply a system that can only be in two states. That is equivalent to say that the observable of interest that is being considered can only have possible values. Is that the case?
If so, a classical bit can have two values...
I was just reading a paper <predatory publisher reference deleted>
There is an argument (originally by Spekkens), in Section 2.1, that is supposed to be against psiontic interpretations. As I understand it, it's that if someone hands you a particle in state x+ or y+ you cannot tell the...
Here is an interesting article off of phys.org that I really liked.
http://phys.org/news/201208caughtcameraquantummechanicsaction.html
What I found interesting is its premise of visually capturing multiple quantum states so that one could personally inspect a lot of these issues that...
Homework Statement
I apologize, this is not really a homework problem. I have an exam coming up, and I need to be able to explain the difference between a stationary/nonstationary quantum state in a qualitative way, and in what cases these states have time dependent probabilities. I am hoping...
Hi,
I recently saw a derivation that included:
[1] #CS = V_spatial * V_momentum
[2] #QS = #CS/h
(where # indicates it's the total number of the variable)
quantum states = QS; classical states = CS; h is Planck's constant
If possible, do you mind explaining or directing me to references...
Homework Statement
Hey, the nocloning theorem states, that arbitrary quantum states cannot be cloned by any circuit.
It is, however, possible to clone orthogonal states.
What would a circuit performing this action look like?
Homework Equations
Relevant equations: I am assuming you all now...
The quantum states ##\psi(x)## of the infinite square well of width ##a## are given by
##\psi(x) = \sqrt{\frac{2}{a}}\sin\Big(\frac{n \pi x}{a}\Big),\ n= 1,2,3, \dots##
Now, I understand ##n \neq 0##, as otherwise ##\psi(x)## is nonnormalisable.
But, can't we get additional states for...
I have to find a unitary transformation that takes me from one quantum state to another (or if there is such a transformation), given the two quantum states in matrix form. The matrices are huge (smallest is 16x16) , so doing it on paper is not an option. Does anyone know how I can do this in...
Hey all,
I'm reading through an anecdotal work about the philosophical foundations of quantum field theory and the authors keep referring to states having the ability to be "sharp." As in it's possible for P to be sharp if the system is mixed, where P is some property of the system. Thanks! IR
Max Tegmark in his paper “Many worlds in context” http://arxiv.org/abs/0905.2182
Argues that …. .“Everett’s MWI is simply standard QM with the collapse postulate removed, so that the Schrödinger equation holds without exception”. He also argues that from this we can deduce that not only...
Homework Statement
A quantummechanical harmonic oscillator with frequency ω has Hamiltonian eigenstates n with eigenvalues En = (n + 1/2) ħω. Initially, the oscillator is in the state (0> + 1>)/√2. Write down how the state of the oscillator evolves as a function of time t. Calculate the...
Here is a thought experiment. Imagine Schrodinger's cat... in the traditional model, there is a single observer outside the box, and the observer creates an entanglement with the catbox device which reveals the quantum superposition of the enclosed cat. The cat is said to be in a superposition...
Hello everyone.
Can someone explains me the meaning of quantum state transition?
For example consider an electron which is in the superposition of two energy eigenstates of a given hamiltonian, now, if no one perturbs the state with a measure, nothing happens and the superposition remains the...
Okay, here goes... Our teacher set a question in the last test which asked us to show that if a system initially be in a stationary state, it will remain in a stationary state even if the system evolves according to the time dependent Schrodinger equation. What I did was show that the...
I am a little confused about exercise 1.2 in the book "Quantum Computation And Quantum Information" By Michael Nielson.
The question is:
Explain how a device which, upon input of one of two nonorthogonal quantum states a> or b> correctly identified the state, could be used to build a device...
Homework Statement
A particle is moving in one dimension, estimate the number of quantum states available to that particle if it is an electron confined in a region 109m long with speed less than 107 m/s (less than meaning velocity is between 107 and 107 m/s)
Homework Equations...
Hello Forum,
When a system is in a particular state, indicated by a A>, we can use any basis of eigenvectors to represent it. Every operator that represents an observable has a set of eigenstates. I bet there are operators with only one eigenstate or no eigenstates. There are operators, like...
Hi guys,
Sorry if this isn't quite the right place to post this, but I have a few conceptual questions that I'd like to clear up about time evolution of a quantum state.
Firstly, what is the exact argument for the evolution operator \hat{U}\left(t,t_{0}\right) being independent of the initial...
I am watching James Binney's QM lectures on iTunes University, and also going through his free textbook. He is a tough teacher, but I love how many misconceptions he points out, and some of the points he makes are very subtle and mind blowing when the lightbulb comes on.
I am confused on...
I am a retired electrical engineer, now able to get back to studying what I really enjoy  mathematics and physics.
As a genuine old geezer, my modern physics knowledge, which was never very deep, is now way out of date. I purchased a copy of "Modern Physics", by Kenneth Krane, and have been...
In any textbooks I have seen, vacuum states are defined as:
a 0>= 0
What is the difference between 0> and 0?
Again, what happens when a+ act on 0> and 0?
and Number Operator a+a act on 0> and 0?
Are systems ever in a pure quantum mechanical state? If they are, is it possible to know the precise pure QM state? The example I am thinking of is the spin of an electron. If we measure the spin about the "zaxis" and find the result to be "up" then we say the electron is in the pure state...
When we speak about wave function of an electron, we write it as ψ_{n,σ} (x,ζ) so that we specify here the orbital quantum number by n and spin quantum number by σ. σ can take two values according to spin up or down. x is space position and ζ has two discrete values related to spin up and down...
Hi,
Just a little thing that's been puzzling me:
Consider a state
\mid \psi \rangle = \frac{1}{\sqrt{2}} \mid A \rangle + \frac{1}{\sqrt{2}} \mid B \rangle
This is normalised since [\frac{1}{\sqrt{2}}]^2 + [\frac{1}{\sqrt{2}}]^2 = 1
Now let A = B:
\mid \psi \rangle =...
I am aware and well read on the decoherence approach to understanding how conglomerations of micro quantum systems will tend to lose quantum coherence via interaction with the environment. The cross terms in the density matrix for the system will tend to zero (due to the partial trace...