User:Wing gundam/Conceptual programs in physics

Different subfields of physics have different programs for determining the state of a physical system.

Classical mechanics

For the simple case of single particle with mass m moving along one dimension x and acted upon by forces $F_{i}$, the program of classical mechanics is to determine the state $x:\mathbb {R} \to \mathbb {R}$ by solving Newton's second law,

$m\,x''(t)=\sum _{i}F_{i}$,

for $x(t)$ given sufficient initial conditions for a second order ordinary differential equation, typically $x(0),x'(0)\in \mathbb {R}$. If these forces are conservative, Newton's law becomes

$m\,x''(t)=\sum _{i}-{\frac {\mathrm {d} U_{i}(x)}{\mathrm {d} x}}$.

In 3 spatial dimensions, the state $\mathbf {x} :\mathbb {R} \to \mathbb {R} ^{3}$ is determined by solving Newton's second law,

$m\,\mathbf {x} ''(t)=\sum _{i}\mathbf {F} _{i}$,

for $\mathbf {x} (t)$ with corresponding initial conditions, typically $\mathbf {x} (0),\mathbf {x} '(0)\in \mathbb {R} ^{3}$. For a system of N particles, Newton's law applies to each particle, constraining an aggregate state $\mathbf {x} :\mathbb {R} \to \mathbb {R} ^{3N}$. Exact solutions exist for many systems of interest, and numerical methods exist for and have been applied to large systems including the pre-solar nebula and planetary atmospheres.

Reformulations

In Lagrangian mechanics for the same system, the state $\mathbf {q} :\mathbb {R} \to \mathbb {R} ^{3N}$ solves Hamilton's principle ${\frac {\delta S}{\delta \mathbf {q} (t)}}=0$ where the action functional is defined as

$S[\mathbf {q} ]\ {\stackrel {\mathrm {def} }{=}}\int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,\mathrm {d} t$.

In Hamiltonian mechanics with canonical coordinates $(\mathbf {q} ,\mathbf {p} )$ and Hamiltonian function ${\mathcal {H}}(\mathbf {q} ,\mathbf {p} ,t)$, the state $\mathbf {q} :\mathbb {R} \to \mathbb {R} ^{3N}$ is determined by solving

$\mathbf {q} '(t)={\frac {\partial {\mathcal {H}}}{\partial \mathbf {p} }}\quad ,\quad \mathbf {p} '(t)=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {q} }}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}={\frac {\partial {\mathcal {L}}}{\partial t}}$.

Quantum mechanics

For a single particle with mass m constrained to the x-axis and subject to a scalar potential $U(x,t)$, the program of quantum mechanics is to determine the wave function $\psi :\mathbb {R} \to L_{2}(\mathbb {R} ^{1},\mathbb {C} )$ where $\psi (x,t)$ solves the Schrödinger equation,

$i\hbar {\frac {\partial }{\partial t}}\psi (x,t)=\left[{\frac {-\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+U(x,t)\right]\psi (x,t)$

given particular initial conditions, for example $\psi (x,0)$ in $L_{2}(\mathbb {R} ^{1},\mathbb {C} )$. Here, $L_{2}(X,E)$ indicates the L² subspace or "square-integrable" subspace of the function space $f:X\to E$. In three dimensions with scalar potential $U(\mathbf {x} ,t)$, the state $\psi :\mathbb {R} \to L_{2}(\mathbb {R} ^{3},\mathbb {C} )$ solves the Schrödinger equation,

$i\hbar {\frac {\partial }{\partial t}}\psi (\mathbf {x} ,t)=\left[{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+U(\mathbf {x} ,t)\right]\psi (\mathbf {x} ,t)$

for corresponding initial conditions, for example $\psi (\mathbf {x} ,0)$ in $L_{2}(\mathbb {R} ^{3},\mathbb {C} )$. Strictly speaking, the space of physically distinct pure states is not the aforementioned L² complex space but rather rays in the projective Hilbert space, which itself stems from the representation theory of C*-algebras. Exact solutions have been found for simple systems like the Hydrogen atom, notably excluding Helium and more complex atoms, while numerical methods exist and have been applied at the molecular level.

Classical limit

The values of the position-space wave function above are the coordinates of the state vector in the position eigenbasis, expressed as $\psi (\mathbf {x} ,t)=\langle \mathbf {x} |\psi (t)\rangle$. The time evolution of the state vector is generated by the Hamiltonian operator ${\hat {H}}$, yielding the general Schrödinger equation $i\hbar {\frac {\mathrm {d} }{\mathrm {d} t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle$, whose formal solution is the unitary time translation operator ${\hat {U}}(t)$,

$|\psi (t)\rangle ={\hat {U}}(t)|\psi (0)\rangle =e^{-{\frac {i}{\hbar }}{\hat {H}}t}|\psi (0)\rangle$.

Expanding the following transition amplitude yields a path integral, taken over all paths $\mathbf {y} :\mathbb {R} \to \mathbb {R} ^{3}$ from $\mathbf {y} (t_{1})=\mathbf {x} _{1}$ to $\mathbf {y} (t_{2})=\mathbf {x} _{2}$,

$\langle \mathbf {x} _{2}|{\hat {U}}(t_{2}-t_{1})|\mathbf {x} _{1}\rangle =\langle \mathbf {x} _{2}|e^{-{\frac {i}{\hbar }}{\hat {H}}(t_{2}-t_{1})}|\mathbf {x} _{1}\rangle =\int _{\mathbf {y} (t_{1})=\mathbf {x_{1}} }^{\mathbf {y} (t_{2})=\mathbf {x} _{2}}e^{{\frac {i}{\hbar }}{\mathcal {S}}[\mathbf {y} ]}\,{\mathcal {D}}\mathbf {y}$,

and convolving this with an initial wave function yields the Lagrangian formulation of quantum mechanics, the path integral formulation,

$\psi (\mathbf {x} _{2},t_{2})=\int _{\mathbf {x} _{1}}\int _{\mathbf {y} (t_{1})=\mathbf {x_{1}} }^{\mathbf {y} (t_{2})=\mathbf {x} _{2}}e^{{\frac {i}{\hbar }}{\mathcal {S}}[\mathbf {y} ]}\,{\mathcal {D}}\mathbf {y} \,\psi (\mathbf {x} _{1},t_{1})\,\mathrm {d} \mathbf {x} _{1}$.

In the limit $\hbar \to 0$ (i.e. as $\hbar /mc$ becomes infinitely smaller than the length scale of interest), the relative contribution of the path $\mathbf {y}$ that solves the classical equations of motion becomes infinite, and consequently ${\hat {U}}(t)$ will transport a decohered wave packet localized at $\mathbf {x} _{1}$ (e.g. $\psi (\mathbf {x} ,t_{1})\approx \delta (\mathbf {x} -\mathbf {x} _{1})$) along its classical path with no quantum effects, generating Hamilton's principle and the program of classical mechanics above.

Quantum field theory

For a field in d spatial dimensions with mass m and value in V, the program of quantum field theory is in theory to obtain the wave functional $\Psi :\mathbb {R} \to L_{2}(\mathbb {R} ^{d}\to V,\mathbb {C} )$ that solves $i\hbar \partial _{0}\Psi [\phi (\cdot ),t]={\hat {H}}\Psi [\phi (\cdot ),t]$ with

${\hat {H}}=\int \mathrm {d} ^{d}x\left[\partial _{0}{\hat {\phi }}(\mathbf {x} ){\hat {\pi }}(\mathbf {x} )-{\mathcal {L}}[{\hat {\phi }},\mathbf {\partial } {\hat {\phi }}]\right]$

given suitable initial conditions, hypothetically $\Psi [\phi (\cdot ),0]:L_{2}(\mathbb {R} ^{d}\to V,\mathbb {C} )$. However, finding an exact solution exceeds current mathematical capabilities for all cases except free particle propagation. In practice, calculations consist of determining scattering amplitudes with perturbative approximations or of numerically approximating corresponding lattice field theories.

Classical limit

The values of the wave functional exist in the field operator's basis as $\Psi [\phi (\cdot ),t]=\langle \phi (\cdot )|\Psi (t)\rangle$, where the state obeys $i\hbar \partial _{0}|\Psi (t)\rangle ={\hat {H}}|\Psi (t)\rangle$. Expanding the formal solution yields a path integral, taken over every field path $\chi :\mathbb {R} \to V^{\mathbb {R} ^{d}}$ from $\chi (t_{1})=\phi _{1}$ to $\chi (t_{2})=\phi _{2}$,

$\langle \phi _{2}|e^{-i{\hat {H}}(t_{2}-t_{1})/\hbar }|\phi _{1}\rangle =\int _{\chi (t_{1})=\phi _{1}}^{\chi (t_{2})=\phi _{2}}e^{{\frac {i}{\hbar }}{\mathcal {S}}[\mathbf {\chi } ]}\,{\mathcal {D}}\mathbf {\chi }$

and convolving this with an initial wave functional yields

$\Psi [\phi _{2},t_{2}]=\int _{\phi _{1}}\int _{\chi (t_{1})=\phi _{1}}^{\chi (t_{2})=\phi _{2}}e^{{\frac {i}{\hbar }}{\mathcal {S}}[\mathbf {\chi } ]}\,{\mathcal {D}}\mathbf {\chi } \,\Psi [\phi _{1},t_{1}]\,{\mathcal {D}}\phi _{1}$.

In the limit $\hbar \to 0$, the relative contribution of the field path $\chi$ that solves the classical equations of field motion dominates, and covariant classical field theory is recovered.

Non-relativistic limit

Every free quantum field ${\hat {\phi }}$ can be decomposed in terms of its annihilation operators as

${\hat {\phi }}(x)={\hat {a}}(x)+{\hat {b}}^{\dagger }(x)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\hbar \omega _{\mathbf {p} }}}}\left\{{\hat {a}}_{\mathbf {p} }e^{-ip\cdot x/\hbar }+{\hat {b}}_{\mathbf {p} }e^{-ip\cdot x/\hbar }\right\}$,

where the momentum space annihilation operators are integrated to yield the operator-valued distributions ${\hat {a}}(x)$ and ${\hat {b}}(x)$, and the energy-momentum relation gives $(\hbar \omega _{\mathbf {p} })^{2}=E^{2}=\left(mc^{2}\right)^{2}+(pc)^{2}$. In the non-relativistic limit $c\to \infty$, this becomes $\hbar \omega _{\mathbf {p} }\approx E\approx mc^{2}$ and the phase $e^{-i\omega _{\mathbf {p} }t}$ and measure $(2\hbar \omega _{\mathbf {p} })^{-1/2}$ factor out, yielding

${\hat {a}}(x)\to {\frac {e^{-imc^{2}t/\hbar }}{\sqrt {2mc^{2}}}}{\hat {A}}(x),\quad {\hat {b}}(x)\to {\frac {e^{-imc^{2}t/\hbar }}{\sqrt {2mc^{2}}}}{\hat {B}}(x)$.

Consequently the field's Lagrangian $L=(\hbar c)^{2}\partial _{a}\phi \partial ^{a}\phi ^{\dagger }-(mc^{2})^{2}\phi \phi ^{\dagger }$ reduces to

$L={\hat {A}}^{\dagger }(i\hbar \partial _{t}+{\frac {\hbar ^{2}\nabla ^{2}}{2m}}){\hat {A}}+{\hat {B}}^{\dagger }(i\hbar \partial _{t}+{\frac {\hbar ^{2}\nabla ^{2}}{2m}}){\hat {B}}+{\text{h.c.}}$

as the annihilation operators dissociate and behave as two separate Schrödinger fields (representing the particle and anti-particle), whose occupied states each independently obey the Schrödinger equation and yield the program of particulate quantum mechanics above.

Other routes

Other routes may encounter issues in defining localized particle states. In the Heisenberg picture and the non-relativistic limit, $e^{imc^{2}t/\hbar }\langle \mathbf {k} |{\hat {\phi }}(x)|0\rangle$ (with $|\mathbf {k} \rangle$ a one-particle state with momentum $\mathbf {k}$) is often identified with a momentum space wave function, but this cannot be localized. When attempting to reduce a relativistic quantum mechanics to non-relativistic quantum mechanics, although the Hamiltonian $H=(\mathbf {p} ^{2}c^{2}+m^{2}c^{4})^{1/2}$ yields the Newton-Wigner propagator and defines a Lorentz scalar $\psi$, unfortunately this propagator is not Lorentz invariant.

References

[[Category:Quantum mechanics]] [[Category:Conceptual modelling]]