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Results tagged with quantum-mechanics
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user 329332
Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy, and the uncertainty principle and is generally used in single-body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.
5
votes
Accepted
Proof of the Variational Theorem
Yes, in general, it is not justified to exchange a limit and an unbounded operator because an unbounded operator is not continuous. However, in your case, you can apply Parseval's identity: If $\{e_n\ …
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4
votes
Accepted
Operator on infinite dimensional Hilbert space: domain and range
The appropriate generalisation of eigenvalues of an operator $A$ on an infinite-dimensional Hilbert space $\mathcal{H}$ is the spectrum $\sigma(A)$ consisting of those $\lambda \in \mathbb{C}$ for whi …
- 668
4
votes
Accepted
Can a set of states be orthonormal, but not complete?
Take as the Hilbert space $\mathbb{C}^2$ and as a set of vectors the set that consists of the vector $(1,0)$ only. This is a set of orthonormal vectors, but it is not complete. It misses a second basi …
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3
votes
Accepted
Do operators always give a number after operating?
An operator $A: V \to W$ is a linear map between two vector spaces $V$ and $W$, e.g. two function spaces. In quantum mechanics, $V$ and $W$ are typically the Hilbert space $L^2(\mathbb{R}^n)$ of squar …
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3
votes
Visual explanation for $|\psi(-\infty)\rangle^{\text{in}}= \lim_{t\rightarrow -\infty} e^{iH...
The idea is the following: In a typical scattering experiment, a particle is shot into some region, interacts there, and eventually leaves this region again. Thus, the interaction is restricted in spa …
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2
votes
In QM, does an algebra containing the Hamiltonian always evolve into itself?
If $\mathcal{A}\subset \mathfrak{B}(\mathcal{H})$ is an algebra of bounded operators on a Hilbert space $\mathcal{H}$ (it is better to restrict to bounded operators because unbounded operators can usu …
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1
vote
Trouble proving properties of the density matrix
Let $\mathcal{H}$ be a finite-dimensional Hilbert space of dimension $n$. The trace of an operator $T$ on $\mathcal{H}$ is defined as
$$\mathrm{Tr}(T) := \sum_{k=1}^n \langle e_k | T |e_k\rangle,$$
wh …
- 668
1
vote
Does this piecewise wave function work for a particle in the box?
The problem is not that much that $\Psi$ is not differentiable in one point. Usually, we consider two functions which are equal almost everywhere (i.e. up to a set of measure 0) to be the same. A sing …
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1
vote
Accepted
Recommendations for Algebraic quantum mechanics book
I can recommend "Araki: Mathematical Theory of Quantum Fields." The first two chapters describe an algebraic formulation of quantum mechanics (and can be read without knowledge of quantum field theory …
1
vote
Why should I not use the Hermitian adjoint in this equation?
Perhaps it becomes clearer with a slightly different notation: If $\psi_p$ is a (generalised) eigenvector of the momentum operator $\hat{p}$ with (generalised) eigenvalue $p$, and $\phi$ another Hilbe …
- 668
0
votes
What is the wave function of a field?
A quantum field $\phi(x)$ (as defined by the Wightman axioms) is a distribution that is operator-valued, that is, if you smear $\phi(x)$ with a smooth rapidly descreasing function $f$, then you obtain …
- 668