In quantum mechanics, the Schrödinger equation describes how a system changes with time. It does this by relating changes in the state of the system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle known. All that remains is to plug the Hamiltonian into the Schrödinger equation and solve for the system state as a function of time.^[1][2]

Often, however, the Schrödinger equation is difficult to solve (even with a computer). Therefore, physicists have developed mathematical techniques to simplify these problems and clarify what is happening physically. One such technique is to apply a unitary transformation to the Hamiltonian. Doing so can result in a simplified version of the Schrödinger equation which nonetheless has the same solution as the original.

Transformation

A unitary transformation (or frame change) can be expressed in terms of a time-dependent Hamiltonian ${\displaystyle H(t)}$ and unitary operator ${\displaystyle U(t)}$. Under this change, the Hamiltonian transforms as:

${\displaystyle H\to UH{U^{\dagger }}+i\hbar \,{{\dot {U}}U^{\dagger }}=:{\breve {H}}\quad \quad (0)}$.

The Schrödinger equation applies to the new Hamiltonian. Solutions to the untransformed and transformed equations are also related by ${\displaystyle U}$. Specifically, if the wave function ${\displaystyle \psi (t)}$ satisfies the original equation, then ${\displaystyle U\psi (t)}$ will satisfy the new equation.^[3]

Derivation

Recall that by the definition of a unitary matrix, ${\displaystyle U^{\dagger }U=1}$. Beginning with the Schrödinger equation,

${\displaystyle {\dot {\psi }}=-{\frac {i}{\hbar }}H\psi }$,

we can therefore insert ${\displaystyle U^{\dagger }U}$ at will. In particular, inserting it after ${\displaystyle H/\hbar }$ and also premultiplying both sides by ${\displaystyle U}$, we get

${\displaystyle U{\dot {\psi }}=-{\frac {i}{\hbar }}\left(UHU^{\dagger }\right)U\psi \quad \quad (1)}$.

Next, note that by the product rule,

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left(U\psi \right)={\dot {U}}\psi +U{\dot {\psi }}}$.

Inserting another ${\displaystyle U^{\dagger }U}$ and rearranging, we get

${\displaystyle U{\dot {\psi }}={\frac {\mathrm {d} }{\mathrm {d} t}}{\Big (}U\psi {\Big )}-{\dot {U}}U^{\dagger }U\psi \quad \quad (2)}$.

Finally, combining (1) and (2) above results in the desired transformation:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\Big (}U\psi {\Big )}=-{\frac {i}{\hbar }}{\Big (}UH{U^{\dagger }}+i\hbar \,{\dot {U}}{U^{\dagger }}{\Big )}{\Big (}U\psi {\Big )}\quad \quad \left(3\right)}$.

If we adopt the notation ${\displaystyle {\breve {\psi }}:=U\psi }$ to describe the transformed wave function, the equations can be written in a clearer form. For instance, ${\displaystyle (3)}$ can be rewritten as

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\breve {\psi }}=-{\frac {i}{\hbar }}{\breve {H}}{\breve {\psi }}\quad \quad \left(4\right)}$,

which can be rewritten in the form of the original Schrödinger equation,

${\displaystyle {\breve {H}}{\breve {\psi }}=i\hbar {\operatorname {d} \!{\breve {\psi }} \over \operatorname {d} \!t}.}$

The original wave function can be recovered as ${\displaystyle \psi =U^{\dagger }{\breve {\psi }}}$.

Relation to the interaction picture

Unitary transformations can be seen as a generalization of the interaction (Dirac) picture. In the latter approach, a Hamiltonian is broken into a time-independent part and a time-dependent part,

${\displaystyle H(t)=H_{0}+V(t)\quad \quad (a)}$.

In this case, the Schrödinger equation becomes

${\displaystyle \stackrel{\dot{}<!--\; \dot{}\; -->}{{\mathrm{\psi <!--\; \psi \; -->}}_I}=-<!--\; -\; -->\frac{i}{\mathrm{\hslash <!--\; \hslash \; -->}}\left(e^{i{H}_{0}t/\mathrm{\hslash <!--\; \hslash \; -->}}V{e}^{-<!--\; -\; -->i{H}_{0}t/\mathrm{\hslash <!--\; \hslash \; -->}}\right)}$Zdroj:https://en.wikipedia.org?pojem=Unitary_transformation_(quantum_mechanics)
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---