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Results tagged with quantum-mechanics
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user 345998
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.
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Value of Measurement in Quantum Mechanics
Consider a Hamiltonian $H$ with discrete eigenvalues $\{E_n\}_{n=1}^\infty$ and eigenstates $\{\psi_n\}_{n=1}^\infty$.
Suppose I prepare a state $\psi=c_1\psi_1+c_2\psi_2$ (normalized) and make a meas …
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answer
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Weyl Ordering of Fermions
Consider Fermion operators $c^\alpha$ and their canonical conjugates $b_{\alpha}$ (satisfying $\{c^\alpha,b_{\beta}\}=\delta^\alpha_\beta$). Is there a prescription for Weyl ordering the following pro …
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Accepted
Weyl Ordering of Fermions
The following excerpt from the paper suggested by Cosmas (Gavazzi '89):
So the answer is that it is the graded average of all permutations of the operators (where graded means we multiply by a minus …
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1
answer
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The classical limit of quantum mechanics through Ehrenfest's theorem
Consider Ehrenfest's theorem:
\begin{align}
m\frac{d\langle x\rangle}{dt}=\langle p\rangle \\
\frac{d\langle p\rangle}{dt}=-\langle V'(x)\rangle.
\end{align}
Suppose $V(x)=x^2+x^{n+1}$ where $n>1$. Th …
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2
answers
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Constraints on Phase Space
This question here motivated me to record to the following fact: Consider a $2n$ dimensional phase space with coordinates $q_1,...,q_n,p_1,...,p_n$. Consider the constraint $C(\vec q)=0$. What is the …
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vote
Accepted
Constraints on Phase Space
First one performs a change of coordinates: $\vec q\to \vec x(\vec q)$. These coordinates are to be chosen such that $x_1(\vec q)=0\longleftrightarrow C(\vec q)=0$.
Let $\vec \pi$ be the momenta conju …
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Why did the Bohr Model Successfully calculate some of the energy levels in hydrogen?
I'm going to answer how the Bohr calculation works. The potential is $V(r)=-Z/r$. The force is $F=Z/r^2$. This has to match the centripetal force $mv^2/r$. So $v=\sqrt\frac{Z}{mr}$. The momentum is $p …
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"Constrain then quantise" vs. "quantise then constrain"
Take your first example of a particle in $\mathbb R^d$ constrained to lie on $x_1=0$. In Dirac Quantization the constraint on the wavefunctions is not $x_1 \psi(\vec x)=0$, but rather $$\frac{\partial …
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answer
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Weinberg's path integral for fermions in Volume 1
In sec. 9.5 of Weinberg's QFT 1, he introduces operators $Q$ and $P$ satisfying
$$
\{Q,P\}=i \tag{9.5.1}
$$
$$
\{Q,Q\}=\{P,P\}=0 \tag{9.5.2}
$$
and eigenstates $|q\rangle$:
$$
Q|q\rangle =q|q\rangle \ …
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