# 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 z-component 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 two-dimensional, constituting a qubit. A pure state here is represented by a two-dimensional 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 semi-definite, 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.

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1. ### I On a constructive method of quantum state preparation (Ballentine)

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...
2. ### I Do black holes destroy quantum states?

This pop article popped up (isn't that what they do, by definition?) on my google news page. https://www.sciencenews.org/article/black-hole-paradoxes-quantum-states It claims that a thought experiment shows that doing a double-slit experiment near a black hole event horizon can reveal...
3. ### I Question on bra vs ket notation

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?
4. ### A Mixed versus Pure Quantum States for the Singlet

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...
5. ### I Is time continuous or discrete in quantum physics?

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...
6. ### I Explaining Quantum States: Global Phase

I was teaching the basics of quantum states and was showing the students that an arbitrary state in a quantum two-level 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...
7. ### Derivation of the density of states?

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...
8. ### I Accuracy of the Density of States

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...
9. ### Engineering Calculating the number of quantum states in the valence and conduction bands

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...
10. ### What are the possible quantum states of an electron relaxing from n=4 to n=3?

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...
11. ### No. of Quantum states available

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...
12. ### I Quantum states: only vectors?

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 complex-valued dimensions, or a restriction to a subspace thereof) , but something else -- say, (higher-order) tensors or bivectors, or some such? My...
13. ### What are energy values and how do they relate to eigenvectors in quantum states?

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...
14. ### Finding the unitary matrix for a beam splitter

Hello, I have some trouble understanding how to construct the matrix for the beam splitter (in a Mach-Zehnder 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...
15. ### Pairwise and joint distinguishability of quantum states

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} =...
16. ### B How are outer products quantum states?

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...
17. ### I Unitary Processes and Completely Identical Quantum States

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...
18. S

See title.
19. ### A Exploring Experimental Techniques for Preparing Quantum States

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...
20. ### I Why is the total number of quantum states = 2n^2 for some n?

If the number of possible values of L is n, and the number of possible values of m is 2*L-1, 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!
21. ### B Constrained Quantum States: Exploring the Meaning

What does it mean to say that a quantum state or wavefunction is constrained?
22. ### B Need Listing of Photon and Particle Quantum States

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...
23. ### I Two quantum states and qubits....

Hello everyone, My understanding is that a two-quantum 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...
24. ### I Psi-epistemic arguments against reality of quantum states

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 psi-ontic interpretations. As I understand it, it's that if someone hands you a particle in state x+ or y+ you cannot tell the...
25. ### I How does a single photon camera pixel work?

Here is an interesting article off of phys.org that I really liked. http://phys.org/news/2012-08-caught-camera-quantum-mechanics-action.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...
26. ### Difference between stationary/non-stationary quantum states

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/non-stationary quantum state in a qualitative way, and in what cases these states have time dependent probabilities. I am hoping...
27. ### Derivation of number of quantum states

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...
28. ### Cloning orthogonal quantum states - circuit?

Homework Statement Hey, the no-cloning 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...
29. ### Additional quantum states of the infinite square well

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 non-normalisable. But, can't we get additional states for...
30. ### Finding a unitary transformation between two quantum states.

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...
31. ### Sharp Quantum States: Exploring Philosophical Foundations

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
32. ### Everett's interpretations and Macroscopic quantum states

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...
33. ### Stupid question about superposition of quantum states

Homework Statement A quantum-mechanical 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...
34. ### Are quantum states relative?

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...
35. ### Quantum states as normal vectors

Are all quantum states represented by normal vectors?
36. ### What happens to an electron's quantum state after measurement?

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...
37. ### Clarification about stationary quantum states of a system

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...
38. ### Understanding Device Identification of Non-Orthogonal Quantum States

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 non-orthogonal quantum states |a> or |b> correctly identified the state, could be used to build a device...
39. ### Calculate number of quantum states of a particle

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 10-9m long with speed less than 107 m/s (less than meaning velocity is between 107 and -107 m/s) Homework Equations...
40. ### Quantum states and representation freedom

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...
41. ### A few conceptual questions on time evolution of quantum states

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...
42. ### Continuous (non-discrete) Quantum States

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...
43. ### Help understanding molecular vs atomic electron quantum states

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...
44. ### Exploring the Possibility of Deducing Quantum States from a Density Matrix

Is it not possible to deduce quantum states from a density matrix?
45. ### Quantum States and ladder operator

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?
46. ### Gaussian Integrals for Quantum States of well Defined Momentum

Consider the Gaussian Integral (eqn 2.64).. is anyone able to explain how the constant of normalization is rationalised?
47. ### Existence of Pure Quantum States

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 "z-axis" and find the result to be "up" then we say the electron is in the pure state...
48. ### Are Spin and Quantum States Independent in Wave Functions?

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...
49. ### Normalisation of quantum states

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 =...
50. ### The arguments against macroscopic quantum states are

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...