- #1
pat1enc3_17
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hey
I got a questions and appreciate obv any reply. I am regarding to Capitel 12, page 128ff. from Gustafson, Sigal; Mathematical Concepts for Quantum Mechanics.
So, let's consider on $L^2(\mathbb R^n)$
\begin{align}
H_n= \sum_{j=1}^n \frac{1}{2 m_j} p_j^2 +V(x)
\end{align}
where $$p_j=-i\hbar\nabla_{x_j},\qquad V(x)=\frac{1}{2} \sum_{i\neq j} V_{ij}(x_i-x_j)$$ is teh momentum operator.
Separation of the centre-of-mass
the latter Hamiltonian has purely essential spectrum, cause it commutes with the total translation of the system
$$ T_h: \psi(x_1,\ldots,x_n) \mapsto \psi(x_1+h,\ldots,x_n+h) $$
First Question: is there a explanation or physical Interpretation behind, why this follows from translationinvariance?
we are now on page 129 middle.
So, now we "break" the translation invariance, with fixing the centre of mass at the origin, to get some interessting spectral information.
Second Question: I don't understand, why we break the system, cause i don't get if we then got some interessting informatino about the spectrum, the old Hamiltonian still have purely essential spectrum. Arent we looking for his Spectrum. I think, I need some hit in the right direction, to understand this.
if further information for the system/ or sth else is needed pls let me know.
bests, tks in adv
pat1enc3_17
I got a questions and appreciate obv any reply. I am regarding to Capitel 12, page 128ff. from Gustafson, Sigal; Mathematical Concepts for Quantum Mechanics.
So, let's consider on $L^2(\mathbb R^n)$
\begin{align}
H_n= \sum_{j=1}^n \frac{1}{2 m_j} p_j^2 +V(x)
\end{align}
where $$p_j=-i\hbar\nabla_{x_j},\qquad V(x)=\frac{1}{2} \sum_{i\neq j} V_{ij}(x_i-x_j)$$ is teh momentum operator.
Separation of the centre-of-mass
the latter Hamiltonian has purely essential spectrum, cause it commutes with the total translation of the system
$$ T_h: \psi(x_1,\ldots,x_n) \mapsto \psi(x_1+h,\ldots,x_n+h) $$
First Question: is there a explanation or physical Interpretation behind, why this follows from translationinvariance?
we are now on page 129 middle.
So, now we "break" the translation invariance, with fixing the centre of mass at the origin, to get some interessting spectral information.
Second Question: I don't understand, why we break the system, cause i don't get if we then got some interessting informatino about the spectrum, the old Hamiltonian still have purely essential spectrum. Arent we looking for his Spectrum. I think, I need some hit in the right direction, to understand this.
if further information for the system/ or sth else is needed pls let me know.
bests, tks in adv
pat1enc3_17