Questions tagged [quantum-states]
The quantum-states tag has no usage guidance.
263
questions
0
votes
2answers
53 views
When is a quantum state pure and when mixed?
Every definition of the two is always very abstract to me. Like, A pure state is located on the surface of the bloch sphere while the mixed state is somewhere within.
First of all, what is an ...
1
vote
1answer
34 views
What information does a quantum state possess?
I am currently studying quantum mechanics and statistical mechanics, and I am rather new to QM. I have a question from the following section in the notes:
Systems of many identical particles, such as ...
0
votes
2answers
48 views
The Negative Energy in the Harmonic Oscillator Potential! [closed]
I'm self studying Quantum mechanics from Griffiths. Now I'm at the Harmonic oscillator potential. All my questions raised after defining the ladder operators $a_-$ and $a_+$.
If $\psi$ satisfies the ...
0
votes
1answer
51 views
Matrix element and expectation value
Can I say that the expectation value of an observable $𝐴̂$ for a state $|𝛼⟩$: $⟨𝐴⟩≡⟨𝛼|𝐴̂|𝛼⟩$ is a more general case of the matrix element$⟨𝛼|𝐴̂|\beta⟩$? I'm not quite clear how are they ...
0
votes
2answers
69 views
Dirac delta function as an inner product
In Shankar's principles of quantum mechanics, the dirac delta function is introduced for generalizing inner products to infinite dimensional spaces. The dirac delta function is such that
$$δ(x-x’) = ⟨...
1
vote
2answers
42 views
The definition of the weaker notion of symmetry in the sense of Wigner's theorem
The weaker notion of symmetry, in the sense of Wigner's theorem, is a transformation on the states that leave all quantum mechanical amplitudes invariant. This tells that such transformations are ...
1
vote
1answer
35 views
The Eigenstates of a Symmetric Operator
Good Afternoon,
By definition, an observable $O$ for a system of N identical particles is symmetric just in case $\langle\psi|O|\psi\rangle = \langle\psi|P^{\dagger}OP|\psi\rangle$ for any permutation ...
-4
votes
1answer
71 views
How to Rigorously Find the Explicit Form of the Eigenvalues Equation?
Even in the simplest case of a free particle, so subjected to the Hamiltonian:
$$H=\frac{\hat{p}^2}{2m}$$
we often need to find the explicit form of the eigenvalue equation:
$$H|E\rangle=E|E\rangle \ \...
1
vote
1answer
88 views
Why do we represent states vectors with ket vectors?
From what I currently understand given a general state vector $|\psi\rangle$ the wave function:
$$\psi(x) = \langle x|\psi\rangle$$
represent the vector $|\psi\rangle$ in the base of the eigenvalues ...
0
votes
0answers
24 views
What represents the state of a quantum system? [duplicate]
Frederic Schuller claims in his lectures that state of a QM system is NOT a vector in a Hilbert Space. Physicists simply make a mistake here. And they make some complementary mistake elsewhere which ...
0
votes
3answers
106 views
About the behavior of the position and momentum operators
Following my book I came to know the following expressions for the position and momentum operators ($\hat{x},\hat{p}$):
\begin{align}&\langle x|\hat{x}|\psi\rangle=x\psi(x) \ \ \ \ \ &(1)\\[1....
3
votes
1answer
81 views
Functions versus Vectors in Quantum Mechanics
In the beginning quantum mechanics is introduced by representing the states as cute little complex vectors, for example:
$$|a\rangle=a_+|a_+\rangle+a_-|a_-\rangle$$
this is a complex vector ...
1
vote
1answer
35 views
The Indistinguishability Postulate
What is often called the Indistinguishability Postulate is expressed in (at least) two different ways depending on the textbook.
For any normalized composite states of N identical particles $|\psi \...
2
votes
1answer
50 views
What is a neutrino state if not a particle?
When reading about the 2015 Nobel prize and how this led to the possibility of the existence of sterile neutrinos I am told that:
"(...) three active neutrinos $\nu_e$, $\nu_\mu$, $\nu_\tau$, are ...
0
votes
1answer
29 views
Exchange Degeneracy of Fermions
Suppose that two identical fermions 1 and 2 are in the following antisymmetric state:
$|\Psi_{1}\rangle = 1/\sqrt{2}(|A\rangle|B\rangle - |B\rangle|A\rangle)$,
where $|A\rangle$ and $|B\rangle$ are ...
-1
votes
1answer
68 views
Bad Notation in Modern Quantum Mechanics by Sakurai
For starters I may be mistaken bringing this up as a notation problem, maybe for some reason unknown to me this is not about notation at all and I simply do not understand the topic well enough yet, ...
2
votes
1answer
48 views
What theories of quantum mechanics can eschew global phase?
Whilst reading the quantum.country essay by Michael Nielsen and Andy Matuschak, they mention (a bit further down from here) that there is formulations of QM that can completely ignore global phase ...
1
vote
1answer
54 views
Relating Fock states to eigenfunctions in space domain
How can I relate the eigenvalues of $H=\hbar\omega(a^\dagger a+1/2)$ to the eigenfunctions of $H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2 x^2$, with $p=-i\hbar\nabla$? I mean, how the analytical approach ...
7
votes
6answers
851 views
Very precisely explaining when phase plays a role or doesn't play a role in QM
My question is probably basic at first view but I would like to really understand this in details.
The way I understand the role of the phase in quantum mechanics is that as soon as we have a physical ...
4
votes
1answer
30 views
Max-relative entropy between a state and its marginals
Background
The quantum relative entropy is defined for any quantum states $\rho, \sigma$ as
$$D(\rho\|\sigma) = tr(\rho\log\rho) - tr(\rho\log\sigma)$$
For arbitrary choice of $\rho,\sigma$, the ...
1
vote
2answers
50 views
Difference between Mixed and Pure states [duplicate]
Suppose that there is a system of two photons 1 and 2, each of which is in a mixed state $1/2|R\rangle\langle R| + 1/2 |L \rangle\langle L|$, where $|R \rangle$ and $\langle L|$ are two orthonormal ...
4
votes
2answers
57 views
Is there a classical analogue to purification of quantum states?
I am trying to understand how quantum states are a generalization of probability distributions and have some issues understanding purifications. A mixed quantum state $\rho_A$ can always be purified ...
0
votes
0answers
39 views
Ket notation in alternate forms
I have been told that I can describe a system by its wave number states:
$$|k_1\rangle|k_2\rangle,$$
and that the following is true:
$$|k_1\rangle|k_2\rangle=|k_1+k_2\rangle|k_1-k_2\rangle,$$
I am ...
0
votes
0answers
19 views
Microstates of Spin-Orbit Levels in Atomic Physics
I have searched for this answer in many atomic physics books and didn't find the answer. I have a question on how to assign the microstates to their associated level. For example, I worked out the ...
0
votes
2answers
37 views
How do you know if the signal is a pure or mixed state when doing state reconstruction (quantum tomography)?
You're trying to reconstruct the density matrix by sampling a signal.
If you measure spins coming from a source along the Z axis, and 50% of the time they are spin up and 50% spin down, how do you ...
2
votes
1answer
69 views
Conservation of Distinctions in Quantum Mechanics
Recently I have been reading Quantum Mechanics The Theoretical Minimum by Leonard Susskind. In the book he mentions the law of conservation of distinctions, i.e. the conservation of information.
He ...
0
votes
1answer
51 views
Angular momentum and Schrodinger equation
I'm studying stationary states and their orbital angular momentum in 3D Schrodinger equation. I have tried to understand by myself the situation but I get lost. I think it might be useful to know ...
1
vote
1answer
33 views
Is a qutrit an example of tripartite entanglement? [closed]
Is a qutrit considered to be in a state of tripartite entanglement?
0
votes
0answers
44 views
Summation of a tensor product of two state functions
In the equation, where the $\vec{\Psi}$'s are particle states,
$$
\sum_{l,m} C_{lm}\Big\{\Big[\frac{n^{2}}{c^{2}}\dfrac{\partial^2}{\partial t^2}\vec{\Psi}_l(\vec{r}_{1},t)-\nabla^{2}_{1}\vec{\Psi}_l(\...
7
votes
2answers
690 views
Bra ket notation rigorous way
I'm struggling to see how $\langle x|\Psi\rangle =\Psi(x)$.
I have read a few previous bra ket questions in here but still not clear.
Any good book for understand the bra-ket notation in more rigorous ...
11
votes
3answers
996 views
What do the quantum fields represent, mathematically?
I am looking for insight on quantum field theory, and more precisely, I am interested in having a low-detailed idea of what a quantum field theory is about; moreover, I should say hat I am a ...
0
votes
2answers
61 views
Finding spin up/spin down eigenstates along some arbitrary direction
So let's say I have some particle in some arbitrary state where its ket vector is given as a linear combination of the spin up eigenstate and the spin down eigenstate. The magnitude square of a, which ...
2
votes
1answer
70 views
Why are time derivatives of states in QFT equal to zero?
In equation 6-38 on page 176 of the book "Student Friendly QFT" by Robert D. Klauber it is said that the partial derivative w.r.t. time of a multi-particle state is equal to zero since we ...
2
votes
2answers
78 views
Fubini Study metric from bra-ket notation
I know that the Fubini Study metric for the SU(2) coherent state is the metric on $CP^1$. The SU(2) un normalised coherent state is given by
$$
\mid z\rangle=\sum_{m=-j}^{m=+j}\sqrt{\left(
\begin{...
0
votes
0answers
30 views
How to tighten the Fuchs-Van de Graaff inequalities for pure and mixed states?
Defining the trace-distance as $D(\rho , \sigma) = \dfrac12 tr|\rho - \sigma|$ and the Fidelity between two quantum states as $F(\rho , \sigma) = tr\sqrt{\sqrt\rho\sigma\sqrt\rho}$
I need to show the ...
0
votes
1answer
52 views
Are density states more fundamental than wavefunctions? [duplicate]
Some interpretations, like the many-worlds interpretation, treat the wavefunction (modulo an overall phase factor) as objective and fundamental.
But consider the following example for a qubit: a ...
2
votes
3answers
153 views
Wave function as a ket vector in a Hilbert space
There's something I don't understand: I've learned that quantum wave functions can be described as a "ket vector" in an abstract vector space called Hilbert space. The position wave function,...
1
vote
2answers
47 views
Entanglement and Mixed States
The Wikipedia page for "Density Matrix" (https://en.wikipedia.org/wiki/Density_matrix) takes each of a pair of entangled photons as an example of a mixed state:
A radioactive decay can emit ...
2
votes
0answers
59 views
Wave function collapse for experimental quantum state preparation
Has the John von Neumann projection, or the wave function collapse, been used in the experimental preparation of quantum states? As an illustration, for a pair of EPR-type entangled 2-level atoms
$$|\...
1
vote
1answer
41 views
What does it mean for a projection operator to represent a state?
I can understand that an idempotent operator can be represented as a projection operator, such as $|x\rangle\langle x|$. But some authors seem to use projection operators, instead of vectors, to ...
0
votes
1answer
38 views
Where does the imaginary unit $i$ come from in representing spin vector along y axis?
I am currently reading Leonard Susskind's - "Quantum Mechanics - The Theoretical Minimum". On Page 38 of the book, the writer described representing spin vectors along the $x$- and $y$-axes ...
4
votes
3answers
124 views
Why is quantum measurement not happening all the time? [duplicate]
I have a question which may be very naive yet I have no answer. I studied undergraduate quantum mechanics 4 years ago now and even if I studied more advanced stuff like QFT I feel like I don't ...
3
votes
1answer
64 views
Why can’t an elementary particle be forced to have a particular outcome in an entangled pair?
I read in a blog Quantum Entanglement: Slower Than Light that one can not force a particle from an EPR pair to have a not statistical outcome for the entanglement parameter. I can not understand why? ...
0
votes
1answer
28 views
Why the coherent state from displacement operator and by expanding in terms of Fock state are not equal?
We can get coherent state from the formula
$$|\alpha\rangle =D(\alpha)|0\rangle = \exp (\alpha a^\dagger-\alpha a)|0\rangle = \exp\left(-\frac{|\alpha |^2}{2} \right) \exp(\alpha a^\dagger) \exp(\...
0
votes
0answers
25 views
Lorentz Transform of field operator and state vector
I'm learning QFT now,I wonder if apply lorentz transform $\Lambda$ to a system, how the state vector $|\psi\rangle$ and field operator $\phi(x)$ will be.
According to some books, the state vector ...
1
vote
0answers
48 views
Can be a Schmidt Decomposition Written in this Form?
The Schmidt Decomposition of a state $|\Psi_{AB}\rangle = \sum C_{ij}|i_A\rangle |j_B\rangle$ can be written in this form $|\Psi_{AB}\rangle = \sum \lambda_k|k_A\rangle |k_B\rangle$ where $\lambda_k$ ...
0
votes
1answer
78 views
How does small discrete steps between rotational energy states make integration accurate?
I understood that if the distance between the discrete rotational energy states are very small, the number of quantum states can be approached accurately using integration of the same formula. This is ...
0
votes
0answers
30 views
Why is preservation of affine structure an assumption for mixed states?
I'm reading Isham's Lecture on Quantum Theory, and towards the end of Chapter 6, in Section 6.4.4, The Time Development of a Mixed State, he shows (not defines) that $\hat{P}_{|\psi_t\rangle}:=|\psi_t\...
0
votes
0answers
32 views
What if classical and quantum probabilities were not independent?
Isham in his Lectures on Quantum Theory Chapter 6, Technical Developments states that for a mixed state $\rho=(\psi_1,\dots,\psi_D;w_1,\dots,w_D)$ and an observable A with an eigenvalue $a_n$, $$\text{...
2
votes
1answer
40 views
Convex combination of product states cannot be purified to a product state
Suppose we have two distinct states $\rho,\sigma\in \mathcal{H}_A$. Define the following state
$$\omega = \frac{1}{2}(\rho^{\otimes n} + \sigma^{\otimes n}) \in \mathcal{H}_A^n$$
Let $\mathcal{H}_R\...
