40
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 ...
- 122k
16
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))...
- 1,012
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 ...
- 69.9k
9
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 ...
- 17.6k
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.
- 46.6k
6
votes
Why does spin acting along $x$ on the spin up state yield spin down?
...What I'm stuggling to understand is why applying $\hat{S}_{x}$ to $\ | \alpha \rangle \ $ yields $\ | \beta \rangle \ $ from an intuitive perspective beyond just plugging in the for $\hat{S}_{x}$ ...
- 17.6k
6
votes
If quantum fields are operator valued distributions, why aren't they always smeared?
The main point of quantum field theory is to study dynamics. The dynamics is given by interactions among the different fields. These interactions are point interactions. The way that these point ...
- 14.2k
5
votes
Dirac's definition of probability in quantum mechanics
Once one identifies $\langle x | A |x \rangle $ with the average value of $A$, and the states $|x\rangle$ are normalized, it is just the frequentist definition of probability that forces to identify ...
5
votes
Accepted
Basis of infinite dimension Hilbert spaces in quantum mechanics
In mathematics, one usually defines the basis of an infinite dimension vector space as a set of vectors such that any other vector can be written as a finite linear combination of the vectors in the ...
- 122k
4
votes
Dirac's definition of probability in quantum mechanics
This has nothing to do with quantum mechanics. Imagine you're tossing a fair die. You can assign any numbers to the six outcomes. If you number them $1,2,3,4,5,6$ then the expected value of a roll is $...
- 25.3k
4
votes
Can $\langle0|(\hat{a}-\hat{b})|0\rangle$ be written as $\langle0|\hat{a}|0\rangle-\langle0|\hat{b}|0\rangle$?
The answer is yes. $$A|0\rangle= (a-b)| 0\rangle = a|0\rangle-b|0\rangle$$
And the scalar product, which is what happens when you combine a ket and bra, is linear. $$\langle0|A|0\rangle= \langle0|(a|0\...
- 41
4
votes
Accepted
If quantum fields are operator valued distributions, why aren't they always smeared?
Yes, if we are mathematically careful about the distributional nature of a quantum field, then we should indeed consistently write quantum fields only as "smeared functions" $\phi(f)$.
But ...
- 122k
4
votes
What experimental proof of quantum superposition do we have?
The answers to your questions are immediate consequences of the definition of a quantum state.
Every quantum state is a superposition for the same reason that every integer is a sum of other integers....
- 14.8k
3
votes
Why is the usage of just linear, unitary operator that isn't necessarily self-adjoint in the middle of a bra-ket a problem?
The point is that the notation is ambiguous. Interpreted as $\langle \beta \lvert A \rvert \alpha \rangle = \langle \beta , A \alpha \rangle$ (where on the right I have reverted to the unambiguous ...
- 6,879
3
votes
Accepted
About sending time to infinity in a slightly imaginary direction in QFT
TL;DR: In the Gell-Mann and Low theorem the $i\epsilon$ prescription in the complex time plane is equivalent to an adiabatic cutoff of interactions. Both serve as a regularization.
In more details: We ...
- 196k
2
votes
Accepted
Trouble understanding Dirac's notation in "The principles of quantum mechanics"
The notation’s purpose is to accommodate for degeneracy. To construct an orthonormal basis, the eigenvalues of the operator are not enough to distinguish the eigenvectors. You need to add some ...
- 9,535
2
votes
Is there an identity for $\hat{a}^+ \cdot \hat{a}^+$?
No!
Note that $a^{\dagger 2}$ is simply the creation operator applied twice. It does not simplify to a unique identity like the commutation relation or like writing $a a^{\dagger}$ as a function of ...
- 1,158
2
votes
Accepted
Problem with understanding the concept of vacuum state of a quantum field
An electric field with a value different than zero in some region of space can be considered as a coherent state of photons Coherent state vs. classical world. This is not the vacuum state. We can ...
- 933
2
votes
Accepted
Inner product not invariant in QM?
The inner product is invariant, and your contradiction comes from an incorrect method of finding bra component vectors the fact that the inner product does not correspond to multiplication of ...
- 775
2
votes
What experimental proof of quantum superposition do we have?
Wave Nature
As many have pointed out, the double-slit experiment is perhaps the canonical demonstration of superposition and a macroscopically visible demonstration of quantum mechanics. But the ...
- 2,328
2
votes
Two-dimensional Two-particle system
You are told both $H_a$ and $H_b$ are simultaneously diagonalizable to diag($E_0,E_1$) in their respective bases. So the full hamiltonian diagonalizes to the 4x4 matrix
$$
H^0= \operatorname{diag}(E_0,...
- 60.4k
2
votes
Projection operator onto support of distinct observables
The space you are looking for is the sum of the two spaces.
The orthogonal projector is denoted by $$P_1\vee P_2$$
where $P_i$ are the projectors on the subspaces.
Actually the wanted space is the ...
- 69.9k
2
votes
Accepted
Identity of bosonic coherent states
The state in the integral is the joint eigenstate of each of the annihilation operators. The annihilation operators are for particular modes labeled here by $x$. I will clarify what I think the ...
- 4,485
2
votes
Is superposition a real quantum state or just an unknown
Is superposition a real quantum state or just an unknown ?
Theories that posit that quantum superposition is a byproduct of our ignorance of (possibly inaccessible) attributes with deterministic ...
- 47.4k
2
votes
Variance/Standard deviation of an observable on a state that is a linear combination of eigenvectors of that observable
The oscillator Hamiltonian is non-degenerate, but this property is not necessary. Without loss of generality, to be self-evident later, stick to a two-dimensional Hilbert space, assuming
$$
A|1\rangle ...
- 60.4k
1
vote
Accepted
Projection operator onto support of distinct observables
This is an answer to the comment. Let $H_1$ and $H_2$ be two closed subspaces of a Hilbert space. Let $H_{1,2} = H_1 + H_2$ be the closure of the algebraic sum of the subspaces. (Note: In finite ...
- 244
1
vote
What experimental proof of quantum superposition do we have?
Now the question: Has it been somehow proven that a physical entity can a some point in time and space have a dual state (independently of the model)?
Science doesn't prove statements. Scientists ...
- 6,622
1
vote
What experimental proof of quantum superposition do we have?
is it only that quantum mechanics is the only known model that allows us to explain things we otherwise couldn't?
Here are two examples from my side.
(1) Quantum dots produce entangled photons in ...
- 10.1k
1
vote
Inner product not invariant in QM?
You said-"This means that any arbitrarily chosen set of basis vectors is guaranteed to be orthonormal!".... It isn't correct, you can have basis vectors which are non orthogonal for instance....
1
vote
Accepted
What does a beam splitter look like in path-encoded notation for at most one photon?
To leave the vacuum unchanged, you need to put a $\sqrt{2}$ in the top left corner.
- 19.7k
Only top scored, non community-wiki answers of a minimum length are eligible