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Lets look at the transition amplitude $U(x_{b},x_{a})$ for a free particle between two points $x_{a}$ and $x_{b}$ in the Feynman path integral formulation

  • $U(x_{b},x_{a}) = \int_{x_{a}}^{x_{b}} \mathcal{D} x e^{\frac{i}{\hbar}S}$

($S$ is the classical action). It is often said that one gets classical mechanics in the limit $\hbar \rightarrow 0$. Then only the classical action is contributing, since the terms with non-classical $S$ cancel each other out because of the heavily oscillating phase. This sounds reasonable.

But when we look at the Heisenberg equation of motion for an operator $A$

  • $\frac{dA}{dt} = \frac{1}{i \hbar} [A,H]$

the limit $\hbar \rightarrow 0$ does not make any sense (in my opinion) and does not reproduce classical mechanics. Basically, the whole procedure of canonical quantization does not make sense:

  • $\{\cdots,\cdots\} \rightarrow \frac{1}{i \hbar} [\cdots,\cdots]$

I don't understand, when $\hbar \rightarrow 0$ gives a reasonable result and when not. The question was hinted at here: Classical limit of quantum mechanicsClassical limit of quantum mechanics. But the discussion was only dealing with one particular example of this transition. Does anyone has more general knowledge about the limit $\hbar \rightarrow 0$?

Lets look at the transition amplitude $U(x_{b},x_{a})$ for a free particle between two points $x_{a}$ and $x_{b}$ in the Feynman path integral formulation

  • $U(x_{b},x_{a}) = \int_{x_{a}}^{x_{b}} \mathcal{D} x e^{\frac{i}{\hbar}S}$

($S$ is the classical action). It is often said that one gets classical mechanics in the limit $\hbar \rightarrow 0$. Then only the classical action is contributing, since the terms with non-classical $S$ cancel each other out because of the heavily oscillating phase. This sounds reasonable.

But when we look at the Heisenberg equation of motion for an operator $A$

  • $\frac{dA}{dt} = \frac{1}{i \hbar} [A,H]$

the limit $\hbar \rightarrow 0$ does not make any sense (in my opinion) and does not reproduce classical mechanics. Basically, the whole procedure of canonical quantization does not make sense:

  • $\{\cdots,\cdots\} \rightarrow \frac{1}{i \hbar} [\cdots,\cdots]$

I don't understand, when $\hbar \rightarrow 0$ gives a reasonable result and when not. The question was hinted at here: Classical limit of quantum mechanics. But the discussion was only dealing with one particular example of this transition. Does anyone has more general knowledge about the limit $\hbar \rightarrow 0$?

Lets look at the transition amplitude $U(x_{b},x_{a})$ for a free particle between two points $x_{a}$ and $x_{b}$ in the Feynman path integral formulation

  • $U(x_{b},x_{a}) = \int_{x_{a}}^{x_{b}} \mathcal{D} x e^{\frac{i}{\hbar}S}$

($S$ is the classical action). It is often said that one gets classical mechanics in the limit $\hbar \rightarrow 0$. Then only the classical action is contributing, since the terms with non-classical $S$ cancel each other out because of the heavily oscillating phase. This sounds reasonable.

But when we look at the Heisenberg equation of motion for an operator $A$

  • $\frac{dA}{dt} = \frac{1}{i \hbar} [A,H]$

the limit $\hbar \rightarrow 0$ does not make any sense (in my opinion) and does not reproduce classical mechanics. Basically, the whole procedure of canonical quantization does not make sense:

  • $\{\cdots,\cdots\} \rightarrow \frac{1}{i \hbar} [\cdots,\cdots]$

I don't understand, when $\hbar \rightarrow 0$ gives a reasonable result and when not. The question was hinted at here: Classical limit of quantum mechanics. But the discussion was only dealing with one particular example of this transition. Does anyone has more general knowledge about the limit $\hbar \rightarrow 0$?

2 corrected formula
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Lets look at the transition amplitude $U(x_{b},x_{a})$ for a free particle between two points $x_{a}$ and $x_{b}$ in the Feynman path integral formulation

  • $U(x_{b},x_{a}) = \int_{x_{a}}^{x_{b}} \mathcal{D} x e^{\frac{i}{\hbar}S}$

($S$ is the classical action). It is often said that one gets classical mechanics in the limit $\hbar \rightarrow 0$. Then only the classical action is contributing, since the terms with non-classical $S$ cancel each other out because of the heavily oscillating phase. This sounds reasonable.

But when we look at the Heisenberg equation of motion for an operator $A$

  • $\frac{dA}{dt} = \frac{i}{\hbar} [H,A]$$\frac{dA}{dt} = \frac{1}{i \hbar} [A,H]$

the limit $\hbar \rightarrow 0$ does not make any sense (in my opinion) and does not reproduce classical mechanics. Basically, the whole procedure of canonical quantization does not make sense:

  • $\{\cdots,\cdots\} \rightarrow \frac{i}{\hbar} [\cdots,\cdots]$$\{\cdots,\cdots\} \rightarrow \frac{1}{i \hbar} [\cdots,\cdots]$

I don't understand, when $\hbar \rightarrow 0$ gives a reasonable result and when not. The question was hinted at here: Classical limit of quantum mechanics. But the discussion was only dealing with one particular example of this transition. Does anyone has more general knowledge about the limit $\hbar \rightarrow 0$?

Lets look at the transition amplitude $U(x_{b},x_{a})$ for a free particle between two points $x_{a}$ and $x_{b}$ in the Feynman path integral formulation

  • $U(x_{b},x_{a}) = \int_{x_{a}}^{x_{b}} \mathcal{D} x e^{\frac{i}{\hbar}S}$

($S$ is the classical action). It is often said that one gets classical mechanics in the limit $\hbar \rightarrow 0$. Then only the classical action is contributing, since the terms with non-classical $S$ cancel each other out because of the heavily oscillating phase. This sounds reasonable.

But when we look at the Heisenberg equation of motion for an operator $A$

  • $\frac{dA}{dt} = \frac{i}{\hbar} [H,A]$

the limit $\hbar \rightarrow 0$ does not make any sense (in my opinion) and does not reproduce classical mechanics. Basically, the whole procedure of canonical quantization does not make sense:

  • $\{\cdots,\cdots\} \rightarrow \frac{i}{\hbar} [\cdots,\cdots]$

I don't understand, when $\hbar \rightarrow 0$ gives a reasonable result and when not. The question was hinted at here: Classical limit of quantum mechanics. But the discussion was only dealing with one particular example of this transition. Does anyone has more general knowledge about the limit $\hbar \rightarrow 0$?

Lets look at the transition amplitude $U(x_{b},x_{a})$ for a free particle between two points $x_{a}$ and $x_{b}$ in the Feynman path integral formulation

  • $U(x_{b},x_{a}) = \int_{x_{a}}^{x_{b}} \mathcal{D} x e^{\frac{i}{\hbar}S}$

($S$ is the classical action). It is often said that one gets classical mechanics in the limit $\hbar \rightarrow 0$. Then only the classical action is contributing, since the terms with non-classical $S$ cancel each other out because of the heavily oscillating phase. This sounds reasonable.

But when we look at the Heisenberg equation of motion for an operator $A$

  • $\frac{dA}{dt} = \frac{1}{i \hbar} [A,H]$

the limit $\hbar \rightarrow 0$ does not make any sense (in my opinion) and does not reproduce classical mechanics. Basically, the whole procedure of canonical quantization does not make sense:

  • $\{\cdots,\cdots\} \rightarrow \frac{1}{i \hbar} [\cdots,\cdots]$

I don't understand, when $\hbar \rightarrow 0$ gives a reasonable result and when not. The question was hinted at here: Classical limit of quantum mechanics. But the discussion was only dealing with one particular example of this transition. Does anyone has more general knowledge about the limit $\hbar \rightarrow 0$?

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When does $\hbar \rightarrow 0$ provide a valid transition from quantum to classcial mechanics? When and why does it fail?

Lets look at the transition amplitude $U(x_{b},x_{a})$ for a free particle between two points $x_{a}$ and $x_{b}$ in the Feynman path integral formulation

  • $U(x_{b},x_{a}) = \int_{x_{a}}^{x_{b}} \mathcal{D} x e^{\frac{i}{\hbar}S}$

($S$ is the classical action). It is often said that one gets classical mechanics in the limit $\hbar \rightarrow 0$. Then only the classical action is contributing, since the terms with non-classical $S$ cancel each other out because of the heavily oscillating phase. This sounds reasonable.

But when we look at the Heisenberg equation of motion for an operator $A$

  • $\frac{dA}{dt} = \frac{i}{\hbar} [H,A]$

the limit $\hbar \rightarrow 0$ does not make any sense (in my opinion) and does not reproduce classical mechanics. Basically, the whole procedure of canonical quantization does not make sense:

  • $\{\cdots,\cdots\} \rightarrow \frac{i}{\hbar} [\cdots,\cdots]$

I don't understand, when $\hbar \rightarrow 0$ gives a reasonable result and when not. The question was hinted at here: Classical limit of quantum mechanics. But the discussion was only dealing with one particular example of this transition. Does anyone has more general knowledge about the limit $\hbar \rightarrow 0$?