3 Minor correction

There are two different views about the semiclassical limit in quantum mechanics, the first is based on a somewhat shaky ground due to the fact that the existence of the Feynman integral is not proved yet. On the other side, Wiener integral, its imaginary time counterpart does exist and one could pretend to work things out from this and then move to the Feynman integral. The other approach relies on substantial mathematical theorems due to Elliott Lieb and Barry Simon in the '70 and is essentially valid for many-body physics. These latter results make the limit $\hbar\rightarrow 0$ and $N\rightarrow\infty$ equivalent while the former is not really a physical limit due to the fact that Planck constant is never zero.

Starting from Feynman path integral, the standard formulation applies to a mechanial problem described from a Lagrangian $L$, normally $L=\frac{\dot x^2}{2}-V(x)$ but one can extend this to more general cases, and then the postulate is that, given a path $x(t)$, this must contribute to the full quantum mechanical amplitude of a particle going from the point $x_a$ to $x_b$ with a term $e^{\frac{i}{\hbar}S}$ being $S=\int_{t_a}^{t_b}dtL(\dot x,x,t)$ the action. All the possible paths contribute and so, the full amplitude will be given by the formal writing $$A(x_a,x_b)\sim\int[dx(t)]e^{\frac{i}{\hbar}\int_{t_a}^{t_b}dtL(\dot x,x,t)}.$$ Be warned that this integral is not proved to exist yet, but the Wiener counterpart, that can be obtained changing $t\rightarrow it$, exists and describes Brownian motion. Now, if you take the formal limit $\hbar\rightarrow 0$ to this integral you will immediately recognize the conditions to apply the stationary phase method to it. This implies that the functional must have an extremum is and this can be obtained by pretending that $$\delta S=\delta \int_{t_a}^{t_b}dtL(\dot x,x,t)=0$$ that is, the paths that give the greatest contribution are the classical ones and one recover the classical limit as a variational principle as learned from standard textbooks.

While this is a quite common approach, to extend what really happens to a macroscopic system that we can see to respect all the laws of classical mechanics, we have to turn our attention to the limit of a large number of particles $N\rightarrow\infty$. In this case one has more rigorous results. These are due to Lieb and Simon as already said above. They published two papers about

Lieb E. H. and Simon B. 1973 Phys. Rev. Lett. 31, 681.

Lieb E. H. and Simon B. 1977 Adv. in Math. 23, 22.

In the first paper, their theorem 4 states

Theorem: For $\lambda < Z$, let $E_N^0$ and $\rho_N^0(x)$ denote the ground-state energy and one-electron distribution function for N spin-$\frac{1}{2}$ electrons obeying the Pauli principle and interacting with $k$ nuclei as described above. Then (a) $N^{-\frac{7}{3}}E_N^0\rightarrow E_1$, as $N\rightarrow\infty$; (b) $N^{-2}\rho_N^0(N^{-\frac{1}{3}}x)\rightarrow\rho_1(x)$ as $N\rightarrow\infty$, where convergence in (b) means that for any domain $D\subset R^3$, the expected fraction of electrons in $N^{-\frac{1}{3}}D$ approaches $\int_D\rho_1 (x)d^3x$.

Where $\rho_1(x)$ and $E_1$ refer to the Thomas-Fermi distribution and the corresponding energy. This theorem states that the limit $N\rightarrow\infty$ for a quantum system, under some mild conditions, is the Thomas-Fermi distribution. A system with this distribution is a classical system. The fact that a system with a Thomas-Fermi distribution is a classical one can be seen through the following two references:

W. Thirring(Ed.), The Stability of Matter: From Atoms to Stars - Selecta of E. Lieb, Springer-Verlag (1997).

L. Hörmander, Comm. Pure. Appl. Math. 32, 359 (1979).

The second paper just gives the mathematical support to derive Thomas-Fermi approximation as the leading order of a classical expansion for $\hbar\rightarrow 0$ that I will not present here.

2 Minor editing

There are two different views about the semiclassical limit in quantum mechanics, the first is based on a somewhat shaky ground due to the fact that the existence of the Feynman integral is not proved yet. On the other side, Wiener integral, its imaginary time counterpart does exist and one could pretend to work things out from this and then move to the Feynman integral. The other approach relies on substantial mathematical theorems due to Elliott Lieb and Barry Simon in the '70 and is essentially valid for many-body physics. These latter results make the limit $\hbar\rightarrow 0$ and $N\rightarrow\infty$ equivalent while the former is not really a physical limit due to the fact that Planck constant is never zero.

Starting from Feynman path integral, the standard formulation applies to a mechanial problem described from a Lagrangian $L$, normally $L=\frac{\dot x^2}{2}-V(x)$ but one can extend this to more general cases, and then the postulate is that, given a path $x(t)$, this must contribute to the full quantum mechanical amplitude of a particle going from the point $x_a$ to $x_b$ with a term $e^{\frac{i}{\hbar}S}$ being $S=\int_{t_a}^{t_b}dtL(\dot x,x,t)$ the action. All the possible paths contribute and so, the full amplitude will be given by the formal writing $$A(x_a,x_b)\sim\int[dx(t)]e^{\frac{i}{\hbar}\int_{t_a}^{t_b}dtL(\dot x,x,t)}.$$ Be warned that this integral is not proved to exist yet, but the Wiener counterpart, that can be obtained changing $t\rightarrow it$, exists and describes Brownian motion. Now, if you take the formal limit $\hbar\rightarrow 0$ to this integral you will immediately recognize the conditions to apply the stationary phase method to it. This implies that the functional must have an extremum is this can be obtained by pretending that $$\delta S=\delta \int_{t_a}^{t_b}dtL(\dot x,x,t)=0$$ that is, the paths that give the greatest contribution are the classical ones and one recover the classical limit as a variational principle as learned from standard textbooks.

While this is a quite common approach, to extend what really happens to a macroscopic system that we can see to respect all the laws of classical mechanics, we have to turn our attention to the limit of a large number of particles $N\rightarrow\infty$. In this case one has more rigorous results. These are due to Lieb and Simon as already said above. They published two papers about

Lieb E. H. and Simon B. 1973 Phys. Rev. Lett. 31, 681.

Lieb E. H. and Simon B. 1977 Adv. in Math. 23, 22.

In the first paper, their theorem 4 states

Theorem: *For $\lambda < Z$, let $E_N^0$ and $\rho_N^0(x)$ denote the ground-state energy and one-electron distribution function for N spin-$\frac{1}{2}$ electrons obeying the Pauli principle and interacting with $k$ nuclei as described above. Then (a) $N^{-\frac{7}{3}}E_N^0\rightarrow E_1$, as $N\rightarrow\infty$; (b) $N^{-2}\rho_N^0(N^{-\frac{1}{3}}x)\rightarrow\rho_1(x)$ as $N\rightarrow\infty$, where convergence in (b) means that for any domain $D\subset R^3$, the expected fraction of electrons in $N^{-\frac{1}{3}}D$ approaches $\int_D\rho_1 (x)d^3x$.*x)d^3x$. Where$\rho_1(x)$and$E_1$refer to the Thomas-Fermi distribution and the corresponding energy. This theorem states that the limit$N\rightarrow\infty$for a quantum system, under some mild conditions, is the Thomas-Fermi distribution. A system with this distribution is a classical system. The fact that a system with a Thomas-Fermi distribution is a classical one can be seen through the following two references: W. Thirring(Ed.), The Stability of Matter: From Atoms to Stars - Selecta of E. Lieb, Springer-Verlag (1997). L. Hörmander, Comm. Pure. Appl. Math. 32, 359 (1979). The second paper just gives the mathematical support to derive Thomas-Fermi approximation as the leading order of a classical expansion for$\hbar\rightarrow 0$that I will not present here. 1 There are two different views about the semiclassical limit in quantum mechanics, the first is based on a somewhat shaky ground due to the fact that the existence of the Feynman integral is not proved yet. On the other side, Wiener integral, its imaginary time counterpart does exist and one could pretend to work things out from this and then move to the Feynman integral. The other approach relies on substantial mathematical theorems due to Elliott Lieb and Barry Simon in the '70 and is essentially valid for many-body physics. These latter results make the limit$\hbar\rightarrow 0$and$N\rightarrow\infty$equivalent while the former is not really a physical limit due to the fact that Planck constant is never zero. Starting from Feynman path integral, the standard formulation applies to a mechanial problem described from a Lagrangian$L$, normally$L=\frac{\dot x^2}{2}-V(x)$but one can extend this to more general cases, and then the postulate is that, given a path$x(t)$, this must contribute to the full quantum mechanical amplitude of a particle going from the point$x_a$to$x_b$with a term$e^{\frac{i}{\hbar}S}$being$S=\int_{t_a}^{t_b}dtL(\dot x,x,t)$the action. All the possible paths contribute and so, the full amplitude will be given by the formal writing $$A(x_a,x_b)\sim\int[dx(t)]e^{\frac{i}{\hbar}\int_{t_a}^{t_b}dtL(\dot x,x,t)}.$$ Be warned that this integral is not proved to exist yet, but the Wiener counterpart, that can be obtained changing$t\rightarrow it$, exists and describes Brownian motion. Now, if you take the formal limit$\hbar\rightarrow 0$to this integral you will immediately recognize the conditions to apply the stationary phase method to it. This implies that the functional must have an extremum is this can be obtained by pretending that $$\delta S=\delta \int_{t_a}^{t_b}dtL(\dot x,x,t)=0$$ that is, the paths that give the greatest contribution are the classical ones and one recover the classical limit as a variational principle as learned from standard textbooks. While this is a quite common approach, to extend what really happens to a macroscopic system that we can see to respect all the laws of classical mechanics, we have to turn our attention to the limit of a large number of particles$N\rightarrow\infty$. In this case one has more rigorous results. These are due to Lieb and Simon as already said above. They published two papers about Lieb E. H. and Simon B. 1973 Phys. Rev. Lett. 31, 681. Lieb E. H. and Simon B. 1977 Adv. in Math. 23, 22. In the first paper, their theorem 4 states Theorem: *For$\lambda < Z$, let$E_N^0$and$\rho_N^0(x)$denote the ground-state energy and one-electron distribution function for N spin-$\frac{1}{2}$electrons obeying the Pauli principle and interacting with$k$nuclei as described above. Then (a)$N^{-\frac{7}{3}}E_N^0\rightarrow E_1$, as$N\rightarrow\infty$; (b)$N^{-2}\rho_N^0(N^{-\frac{1}{3}}x)\rightarrow\rho_1(x)$as$N\rightarrow\infty$, where convergence in (b) means that for any domain$D\subset R^3$, the expected fraction of electrons in$N^{-\frac{1}{3}}D$approaches$\int_D\rho_1 (x)d^3x$.* Where$\rho_1(x)$and$E_1$refer to the Thomas-Fermi distribution and the corresponding energy. This theorem states that the limit$N\rightarrow\infty$for a quantum system, under some mild conditions, is the Thomas-Fermi distribution. A system with this distribution is a classical system. The fact that a system with a Thomas-Fermi distribution is a classical one can be seen through the following two references: W. Thirring(Ed.), The Stability of Matter: From Atoms to Stars - Selecta of E. Lieb, Springer-Verlag (1997). L. Hörmander, Comm. Pure. Appl. Math. 32, 359 (1979). The second paper just gives the mathematical support to derive Thomas-Fermi approximation as the leading order of a classical expansion for$\hbar\rightarrow 0\$ that I will not present here.