Editing Virial theorem (section)

== In quantum mechanics ==

Although originally derived for classical mechanics, the virial theorem also holds for quantum mechanics, as first shown by [[Vladimir Fock]]<ref>{{cite journal | last = Fock | first = V. | s2cid = 122502103 | year = 1930 | title = Bemerkung zum Virialsatz | journal = Zeitschrift für Physik A | volume = 63 | issue = 11 | pages = 855–858 | doi = 10.1007/BF01339281|bibcode = 1930ZPhy...63..855F }}</ref> using the [[Ehrenfest theorem]].

Evaluate the [[commutator]] of the [[Hamiltonian (quantum mechanics)|Hamiltonian]]
<math display="block">
 H = V\bigl(\{X_i\}\bigr) + \sum_n \frac{P_n^2}{2m_n}
</math>
with the position operator {{mvar|X<sub>n</sub>}} and the momentum operator
<math display="block">
 P_n = -i\hbar \frac{d}{dX_n}
</math>
of particle {{mvar|n}},
<math display="block">
 [H, X_n P_n] = X_n [H, P_n] + [H, X_n] P_n = i\hbar X_n \frac{dV}{dX_n} - i\hbar\frac{P_n^2}{m_n}.
</math>

Summing over all particles, one finds that for
<math display="block">
 Q = \sum_n X_n P_n
</math>
the commutator is
<math display="block">
 \frac{i}{\hbar} [H, Q] = 2 T - \sum_n X_n \frac{dV}{dX_n},
</math>
where <math display="inline">T = \sum_n P_n^2/2m_n</math> is the kinetic energy. The left-hand side of this equation is just {{math|''dQ''/''dt''}}, according to the [[Heisenberg equation]] of motion. The expectation value {{math|{{angbr|''dQ''/''dt''}}}} of this time derivative vanishes in a stationary state, leading to the '''''quantum virial theorem''''':
<math display="block">
 2\langle T\rangle = \sum_n \left\langle X_n \frac{dV}{dX_n}\right\rangle.
</math>

=== Pokhozhaev's identity ===
{{Unreferenced section|date=April 2020}}
In the field of quantum mechanics, there exists another form of the virial theorem, applicable to localized solutions to the stationary [[nonlinear Schrödinger equation]] or [[Klein–Gordon equation]], is [[Pokhozhaev's identity]],<ref>{{Cite journal |last1=Berestycki |first1=H. |last2=Lions |first2=P.-L. |year=1983 |title=Nonlinear scalar field equations, I existence of a ground state |url=https://link.springer.com/article/10.1007/BF00250555 |journal=Arch. Rational Mech. Anal. |volume=82 |issue=4 |pages=313&ndash;345 |doi=10.1007/BF00250555 |bibcode=1983ArRMA..82..313B |s2cid=123081616 }}</ref> also known as [[Derrick's theorem]].
Let <math>g(s)</math> be continuous and real-valued, with <math>g(0) = 0</math>.

Denote <math display="inline">G(s) = \int_0^s g(t)\,dt</math>.
Let
<math display="block">
 u \in L^\infty_{\text{loc}}(\R^n), \quad
 \nabla u \in L^2(\R^n), \quad
 G(u(\cdot)) \in L^1(\R^n), \quad
 n \in \N
</math>
be a solution to the equation
<math display="block">
 -\nabla^2 u = g(u),
</math>
in the sense of [[Distribution (mathematics)|distributions]].
Then <math>u</math> satisfies the relation
<math display="block">
 \left(\frac{n - 2}{2}\right) \int_{\R^n} |\nabla u(x)|^2 \,dx = n \int_{\R^n} G\big(u(x)\big) \,dx.
</math>