44
votes
What experimental proof of quantum superposition do we have?
"Being in superposition" is not an objective property of a quantum mechanical state. Quantum mechanical states live in a Hilbert space, where, since it is a vector space, every state can be ...
17
votes
What experimental proof of quantum superposition do we have?
I guess you might know that if you have a linear equation $\mathcal{L}$ and two solutions of it, then a superposition of these solutions is also a solution of this linear equation.
$$
\mathcal{L}(f(x))...
16
votes
Accepted
What is the Hilbert dimension of a Fock space?
You need to distinguish two different notions of isomorphism here: An isomorphism of Hilbert spaces and an isomorphism of representations of algebras on Hilbert spaces.
All Hilbert spaces of the same ...
16
votes
Accepted
How to rotate an electron mathematically?
(I wrote this answer in a hurry yesterday, I now fixed a huge number of typos and various mistakes, sorry.)
Well, there are a number of important issues in the question.
First of all, pure quantum ...
14
votes
Are a Hilbert space's dimensions physical?
There’s no relation between the dimension of the Hilbert space and the actual physical space. If you work with a system of angular momentum $2$ - say hydrogen states with $n=3$ and $L=2$ - then your ...
14
votes
If quantum fields are operator valued distributions, why aren't they always smeared?
Yes, the quantum fields must be smeared in order to become well-behaved (symmetric, densely defined) operators (in the Hilbert space of the theory). In mathematically-minded textbooks it is the ...
12
votes
Accepted
What does sandwiching with an unitary operator and its inverse imply?
This goes back to the paradigm that if a ket transforms as $|\psi\rangle\to \hat{U}|\psi\rangle$ under a unitary symmetry transformation $\hat{U}$, then a bra transforms as $\langle\phi|\to \langle\...
12
votes
Accepted
Some questions about derivation of uncertainty principle
It is true that by taking step (2) we are 'unnecessarily' making the uncertainty principle weaker. Furthermore, it's possible to use $|z|^2=Re(z)^2+Im(z)^2$ to derive the stronger Robertson-...
11
votes
What physical processes other than scattering are accounted for by QFT? How do they fit into the general formalism?
We have a model (the standard model) that is formulated as a particular QFT, that seems to describe every experiment we've ever done on Earth (excluding anything gravitational). My point being that ...
11
votes
Accepted
Dirac's definition of probability in quantum mechanics
I'm currently reading "The principles of quantum mechanics" by Dirac...
...What I don't get is the following part, where he writes:
In the general case we cannot speak of an observable ...
10
votes
Accepted
Eigenvalues of position operator in higher dimensions is vector, not scalar?
If you want to model a measurement as a hermitian operator, you must assign distinct real numbers to the possible outcomes, whether they naturally correspond to real numbers or not. This is a ...
10
votes
Physical interpretation of the inner product between two quantum states
Assume the the particle is in state $\psi_B\,.$ The projection
$$
P_A=|\psi_A\rangle\langle\psi_A|
$$
is a Hermitian operator having eigen values zero and one which can be used to measure if the ...
9
votes
Eigenvalues of position operator in higher dimensions is vector, not scalar?
Don't overthink it - the simplest way to think about the vector of position operators $\hat{\vec x}$ is just as notation: When someone writes that $\hat{\vec x} \lvert \vec x_0\rangle = \vec x_0\lvert ...
9
votes
States created by local unitaries in QFT
According to the Reeh-Schlieder theorem (assuming quite standard hypotheses on your QFT in Minkowski spacetime) the subspace of vectors you consider is dense in the whole Hilbert space. Therefore ...
9
votes
Accepted
Determining Bound States from Møller Operator
If $H=H_0+V$, where $V$ is a so-called Kato potential, then the point spectrum of $H$ corresponds to the bound states, while the scattering states correspond to the continuous spectrum (Ruelle's ...
8
votes
Accepted
What exactly is the relation of the continuous spectra with intervals?
No, the continuous spectrum can also be just a single point, which is the case for the operator $H^{-1}$, where $H$ is the Hamiltonian of the harmonic oscillator, as explained here. More generally, ...
8
votes
Position and momentum basis
One postulate of QM is that quantum states live in Hilbert space, and thus must be normalisable.
Another postulate of QM is that if you have a physical observable, then it is representable by a ...
8
votes
Physical interpretation of the inner product between two quantum states
Imagine you have a coherent beam of light / electrons / cold atoms incident on two slits, $A$ and $B$. First, you cover $B$ and measure the wave field $I_A(r) = \langle \psi_A (r)| \psi_A(r) \rangle$. ...
8
votes
Accepted
Confusion about outer product in QM
The quantity $|a\rangle\langle b|$ is a linear operator; it sends kets to kets. Linear operators are equivalent to matrices (if the Hilbert space if finite dimensional). This can be seen by writing it ...
8
votes
What is the Majorana stellar representation?
The Majorana stellar representation is a way to geometrically visualize pure spin-s states. In essence, the Majorana stellar representation 1) establishes a bijection between states of Hilbert space ...
8
votes
Accepted
When do two state functions represent the same quantum state?
The set of wavefunctions which corresponds to the same state as $\psi$ is just the set of multiples of $\psi$ by a non-zero complex number (respectively, a complex number of absolute value $1$, if you ...
8
votes
How to rotate an electron mathematically?
This experiment has actually been done with neutrons. The first publication is Werner et al., PRL 35, 1053 (1975).
The experiment was done with a neutron interferometer. A low-intensity beam of ...
rob♦
- 89.9k
7
votes
Accepted
Calculating eigenvalues and eigenstates of an infinite dimensional Hamiltonian
The following procedure is only on a formal level. Let $k\in [-\pi,\pi)$ and define $$|k\rangle :=\frac{1}{\sqrt{2\pi}}\sum\limits_{m\in\mathbb Z} e^{ikm}|m\rangle \tag 1 \quad .$$
Note that $|k\...
7
votes
Accepted
How to find the density operator of two joint systems given the density operator of the individual systems?
This is not possible in general, as both density matrices could be the reduced density matrices of several joint density operators, see e.g. here and here.
The reduced density matrices determine the ...
7
votes
Is there a generalization of the adiabatic theorem into a degenerate Hamiltonian?
Yes there is. Naturally these degeneracies should be motivated by symmetry considerations or else they would typically destroyed by perturbations. In particular, to have non trivial irreducible ...
7
votes
Accepted
Trace of operators
In complex Hilbert spaces, one of the various equivalent definitions of trace class operator is a bounded operator $T: H\to H$ such that, for every Hilbert basis $\{u_a\}_{a\in A}$, the sum $$\sum_{a\...
7
votes
Overlap between eigenstates of angular momentum operators
The transformation that takes you from $S_z$ to $S_x$ is a rotation about $\hat y$ by $\pi/2$ so all eigenstates of $S_x$ are of the form
$$
\vert jm\rangle_x=R^{-1}_y(\pi/2)\vert jm\rangle_z
$$
so ...
7
votes
Confusion about outer product in QM
Probably the easiest way to see what the outer product is to act with it on a state on our Hilbert space. Define the Hilbert space $\mathcal{H}$ and an orthonormal basis on this vector space $\{|e_{i}\...
7
votes
Accepted
Problem deducing Ehrenfest's theorem using Schrödinger's equation
This is because you do not have to take the complex conjugate, you have to take the adjoint:
$$
\frac{\partial}{\partial t}|\psi\rangle=-\frac{i}{\hbar}\hat{H}|\psi\rangle \ ,
$$
$$
\frac{\partial}{\...
7
votes
Accepted
Can $\langle0|(\hat{a}-\hat{b})|0\rangle$ be written as $\langle0|\hat{a}|0\rangle-\langle0|\hat{b}|0\rangle$?
Yes, this follows from the definition of subtraction of linear operators. $(a-b)|0\rangle = a |0\rangle - b |0\rangle$ and likewise for the bras.
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