Questions tagged [quantum-states]
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346
questions
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0answers
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Creating orthogonal quantum states from a set of given (possibly linearly independent) quantum states
I want to understand how to orthogonalize a system of qubits.
Suppose I have $n$ sets of quantum states like
$$\{ |1_i\rangle|2_i\rangle|3_i\rangle \cdots|k_i\rangle \mid i=1 \dots n \}$$ where $i=1, \...
0
votes
1answer
39 views
Normalization of momentum eigenstates in QFT
Inspired by a previous question, I'd like to ask about the normalization of one-particle states in QFT.
The most common normalization seems to be the covariant one:
$$ \langle \vec p'|\vec p\rangle = (...
0
votes
1answer
40 views
How the normalization condition implies the following relation?
Using equation 2.35 from Peskin and Schroeder:
$$
|\vec{p}\rangle=\sqrt{2 E_{\vec{p}}} a^{\dagger}_\vec{p} |0\rangle
$$
should lead to
$$
U(\Lambda)|\vec{p}\rangle = |\Lambda \vec{p}\rangle,
$$
where ...
0
votes
1answer
18 views
Quark state representation
Quark can be described by irreducible state representation
$\left|(R)Y,I,I_{3}\right>$.
What means these symbols?
I found example
$$\left|u\right>=\left|(3)\frac{1}{3},\frac{1}{2},\frac{1}{2}\...
0
votes
1answer
36 views
Reducing the number of parameters of a quantum state from 4 to 3
We have a quantum state
$$
|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,
$$
where $\alpha$ and $\beta$ are complex numbers, i.e. $\alpha = a + bi$ and $\beta = c + di$. Therefore, our current ...
2
votes
2answers
79 views
What are the relations between mixed/pure/separable/entangled states?
I read this question which clarified the concepts of mixed/pure/separable/entangled states, but I can't see if one of "mixed/pure" implies one of "separable/entangled" and/or vice ...
-1
votes
1answer
59 views
Is there any importance to notations like $| \Omega V\rangle$, $|aV\rangle$?
I'm not really liking this notation. Before this notation, neither of bras and kets have any preference over the other. Either of $|V\rangle$ and $\langle V|$ can be understood as the adjoint of the ...
0
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0answers
37 views
Can we maintain a standing wave in a slowly growing box?
I believed that the answer to that simple question was 'yes', until I was thinking more about it recently, and now I have a doubt.
Consider a simple standing wave in a 1D "box" of length $...
2
votes
1answer
49 views
How to translate from a state/density matrix formalism to matrix product state representation?
From what I understand, MPS is just a simpler way to write out a state, compared to the density matrix. But how do I get those $A_i$ matrices? From all the examples I read, people just somehow "...
2
votes
5answers
378 views
Eigenstates/vectors of the sum of non-commuting Hamiltonians
Suppose I have two Hamiltonians $H_1$ and $H_2$, and they're both two-level systems (they do not commute, such as pauli $X$ and $Z$). $H_1$ has eigenstates $|\psi_{11}\rangle$ and $|\psi_{12}\rangle$, ...
3
votes
1answer
127 views
Why quantum mechanics uses density operators instead of probability distributions over state space?
Whenever I try to get my head around mixed states I am referred to the notion of density operators. I think that density operators were introduced to represent mixed states as operators.
For what I ...
0
votes
2answers
42 views
Parameterizing a unitary transformation between pure states and viewing it as a rotation [closed]
Let $\vert\psi\rangle$ and $\vert\phi\rangle$ be pure states. Then there exists some unitary $U_t$ such that $U_t\vert\psi\rangle = \vert\phi\rangle$.
I have a geometric picture in my mind but I'm not ...
1
vote
2answers
59 views
Why $\frac{1}{E-H_0+i\eta}|n\rangle \stackrel{?}{=} \frac{1}{E-E_n+i\eta}|n\rangle$?
A Hamiltonian $H_0$ is diagonalized in $\{|n\rangle\}$ i.e. $H_0|n\rangle=E_n|n\rangle$. Why can we write
$$\frac{1}{E-H_0+i\eta}|n\rangle \stackrel{?}{=} \frac{1}{E-E_n+i\eta}|n\rangle$$
$H_0$ is in ...
-2
votes
1answer
44 views
Energy Eigenstate of a Hamiltonian
What exactly are energy eigenstates?
For something like H = $h\omega \left( \begin{matrix}1 & 2i \\-2i & 4
\end{matrix}\right)$ , what the eigenstates be like the eigenvectors, so $(i\,\,\,2)$ ...
1
vote
3answers
120 views
What is the meaning of the ket states in the notation $\langle x_f,t_f|x_i,t_i\rangle$?
Path-integral amplitudes are denoted by the inner product $\langle x_f,t_f|x_i,t_i\rangle$ where $|x_i,t_i\rangle$ is a time-independent position eigenstate of the time-dependent Heisenberg picture ...
4
votes
1answer
154 views
Is unphysical states still unphysical in an interaction theory?
Maggiore A modern introduction to quantum field theory Section 4
: In free quantum electromagnetic field theory... since only the two degree of freedom transverse wave, the energy, and the momentum, ...
1
vote
0answers
55 views
QFT - Bra-ket notation and correct operator interpretation
I'm taking a course in Quantum Field Theory and I'm having hard times understanding the notation adopted. Let me be more precise. Considering the Lagrangian for the (classical) real scalar field [...
1
vote
2answers
95 views
Difference between normalized wave function and orthogonal wave function?
I am confused about when we say that a wave function is NORMALIZED so that we say that the indefinite integral of the wave function squared = 1, vs. when we say that the wave function represents ...
0
votes
0answers
24 views
Quantum numbers for diatomic molecules
A diatomic molecule:
is always symmetric if we rotate it along its internuclear axis. Let it be the z-axis.
Then we can say that $[{\cal \hat H}, {\hat L}_z] =0 $, so there is a common set of ...
0
votes
1answer
40 views
Eigenvalue of a vector in a subspace [closed]
Consider, a quantum system has a hamiltonian with eigenstates $\{|\phi_1\rangle,|\phi_2\rangle,|\phi_3\rangle\}$ and associated eigenvalues $\{\lambda_a,\lambda_a,\lambda_b\}$. My notes state that any ...
0
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1answer
51 views
Understanding how to determine the time evolved state vector for a unitary operator constructed from non-commutating operators
Suppose we have a time independent hamiltonian $$H = \hbar g (\sigma_x + \sigma_y + \sigma_z)$$
I know that the unitary operator is as follows:
$$U(t) = exp(-iHt/{\hbar})$$
Sinnce the pauli spin ...
0
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0answers
37 views
Can we find eigenvalues of an operator without knowing the boundary conditions?
Imagine I have a space spanned by the two basis vectors $e_1 = sin x$ and $e_2=cos x$ and I define some inner product using $\langle f|g\rangle = \frac{1}{π}\int_0^{2\pi} f^*g dx$ and some operator $D ...
-3
votes
1answer
65 views
Why does $ O = |\phi\rangle \langle\psi|$ equal $O =\lambda |\phi\rangle \langle\phi|\psi\rangle\langle\psi|$ for the 2 vectors of Hilbert space [closed]
If we take the operator $$\hat{O} = |\phi\rangle \langle\psi| \space \space(1)$$ whereby $|\phi\rangle$ and $|\psi\rangle$ are two vectors of the hilbert space.
My notes also state that $\hat{O}$ can ...
0
votes
2answers
73 views
How does the tranpose conjugate of an operator act on a bra and a ket in the context of annihilation and raising operators?
Consider the annihilation and raising operators as follows:
$$\hat a|n\rangle=\sqrt{n}|n-1\rangle\qquad\text{and}\qquad\hat a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle$$
I know normally if I have an ...
1
vote
2answers
68 views
Reconstructing wavefunction from the density matrix
Say I have a state, $$| \Psi \rangle = \frac{1}{\sqrt 2} \left( | 0 \rangle + \exp( \text{i} \phi ) | 1 \rangle \right) = c_{0} | 0 \rangle + c_{1} | 1 \rangle.$$
Now I construct the density matrix (...
3
votes
2answers
98 views
Examples of Quantum Superposition for the layman?
I'm trying to introduce Quantum Superposition to high-schoolers, and I feel like it would be nice to start with some real-world examples of where it comes up. I'm hoping for non-contrived examples ...
1
vote
3answers
69 views
Why don't we observe superpositions?
Say we are doing the Stern-Gerlach experiment. Here's my understanding of what decoherence tells us. The particle starts in a superposition of spin up and spin down, but then gets entangled with the ...
1
vote
1answer
82 views
Does the ket $\mid \psi \rangle$ and the ket $ \alpha \mid \psi \rangle$ represent the same state?
To show that the state spin $\frac{1}{2}$ particles get a minus sign under a $2\pi$ rotation the following experiment is made
A beam of neutron is divided in two beams $A$ and $B$. The beam $A$ ...
0
votes
0answers
20 views
Mixed States, Gibbs States, and Temperature
I am trying to help myself clarify some concepts. The purpose of this post is twofold: 1) to get some comments/feedback, and 2) to help me organize my thoughts. I have been approaching these concepts ...
0
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0answers
22 views
Does quantum probability entangle reliability?
When I search here for "probabilty reliability" I find eight questions, only a couple coming close to but not exactly on this one. Perhaps this has been covered elsewhere but mild googling ...
2
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1answer
76 views
Question regarding product state and Schrödinger equation
Assume we look at the state $|\Psi(t) \rangle = |\psi(t)\rangle |m(t)\rangle$ with the Hamiltonian acting in the two Hilbert spaces for example: $$\hat{H} = \frac{\hat{p}^2}{2M} \otimes (| \uparrow \...
3
votes
1answer
86 views
Is there an alternative to Fock Space and Hilbert Space in quantum field theory? [duplicate]
Why were Fock Space and Hilbert Space used in quantum field theory?
What was the motivation for choosing them over other mathematical techniques?
0
votes
2answers
123 views
Eigenvalues and eigenstates of $\hat{L}=\sqrt{\hat{L}^2}=\sqrt{\hat{L}^2_x+\hat{L}^2_y+\hat{L}^2_z}$ (without squaring)
I know that $$ \hat{L}^2 \left| l,m \right> = \hbar^2 l (l+1) \left| l,m \right> .$$ Does this mean that $$ \hat{L} \left| l,m \right> = \hbar \sqrt{l (l+1)} \left| l,m \right> ? $$ If so, ...
0
votes
2answers
86 views
Hadamard gate over 2 qubits [closed]
Let H be the Hadamard gate:
$$(\frac{1}{\sqrt{2}})\begin{pmatrix}\begin{array}{rrrrrrrr} 1 & 1 \\ 1 & -1 \end{array}\end{pmatrix}$$
I would like to write down the matrix associated to the ...
-2
votes
1answer
62 views
How to determine if a state is entangled or pure in general?
Given a state like $a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$ where $a,b,c,d$ satisfy normalization, how can we know if it's pure?
0
votes
1answer
95 views
What state does a system collapse to after measuring a degenerate eigenvalue?
$\newcommand{\ket}[1]{|#1\rangle}$
Let $\hat A$ be some observable, and $\ket n$ and $\ket m$ two degenerate eigenstates with eigenvalue $a$, such that
$$\hat A \ket n=a\ket n,$$
$$\hat A \ket m=a\ket ...
0
votes
1answer
58 views
Does a 2D (real-space) wavefunction always has to be a product of two 1D wavefunction (i.e. always separable)?
In the first-quantization formalism for many particle quantum mechanics, let $|x \rangle$ and $|y \rangle$ be two basis for two particles $A$ and $B$: $\psi_A(x) = \langle x | \psi_A \rangle$ and $\...
-2
votes
1answer
64 views
Doubt about the Gaussian state
I am reading an article that makes an application using the Gaussian state. The author of the article writes the Gaussian state as follows:
$$\psi(q) = [2\pi(\Delta q)^2]^{-\frac{1}{4}}e^{-\frac{q^2}{...
1
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0answers
20 views
Interpretation of measurements and Eigenstates for continuous variables [duplicate]
I find myself (probably like many others) somewhat unclear on the implications of the postulate of quantum mechanics that "measurements of a value leave the system in an eigenstate". The ...
8
votes
5answers
1k views
What are the necessary and sufficient conditions for a wavefunction to be physically possible?
Often times it is stated in books that a quantum state is physically realizable only if it is square integrable. For example in Griffiths (2018 edition) page 14 he stated
Physically realizable states ...
5
votes
2answers
185 views
Hilbert space in Dirac's representation
I've been reading some old posts here on physics stack exchange and I realized something that have never ocurred to me before.
Let $\mathscr{H}$ be a Hilbert space over $\mathbb{C}$. An orthonormal ...
1
vote
3answers
161 views
What is the significance of a Hilbert space? [closed]
I see the term Hilbert space thrown around a lot. At first I thought it is just when you have complex vectors and define an inner product between them. However, it seems to be a lot more than that ...
5
votes
6answers
768 views
Is a wave function a ket?
I just started with Dirac notation, and I am a bit clueless to say the least. I can see Schrödinger's equation is given in terms of kets. Would I be correct to assume if I were given a wavefunction, ...
1
vote
0answers
53 views
Degenerate observable in quantum mechanics
An observable in quantum mechanics is represented by a hermitian matrix. When an observation is made the state collapses to one of the eigenstates of the matrix. What happens if some eigenvalues are ...
3
votes
2answers
116 views
Riesz Representation Theorem and Inner Products
I am running into some confusion:
Suppose in Quantum Mechanics we think of the ket $|x \rangle $ as being an improper state - one not "actually" in the Hilbert space, but which is useful to ...
-3
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1answer
70 views
In quantum mechanics, do quantum systems not exist before measurement? [closed]
If a quantum system manifests itself only during measurement, and if the wave function is only our knowledge of a quantum system, maybe it does not physically exist at all before measurement? Maybe a ...
0
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0answers
34 views
In and out states, QFT [duplicate]
In s matrix calculation we use in and out states. I have few doubts about in and out states.
In which representation do we discuss about in and out states (Heisenberg picture or interaction picture)?
...
-1
votes
1answer
72 views
Is the existence of absolutely separable quantum states possible in nature? [closed]
Can quantum systems be in an absolutely separable state, without any relationship? Or does this approximation and some measure of entanglement (for example, through gravitational interaction) still ...
0
votes
3answers
78 views
Relations between Dirac-notation representation of wave function and wave function in a particular basis
My lecturer has emphasised on a number of occasions that:
$$\ |\psi \rangle \neq | \psi(p) \rangle \label{a}\tag{1}$$
since $$\psi(p) = \langle p| \psi \rangle = \int \langle p |r \rangle \langle r | ...
0
votes
1answer
53 views
Why does the hamiltonian in the operator act on the wave function?
Suppose we have the following relation:
$H_0 | \phi_1\rangle = E_1 |\phi_1 \rangle $
Why is it that if we take the unitary function $$U_{0} = \exp\left(\frac{-iH_{0}t}{\hbar}\right)$$
and apply it to ...
