Quantum reference frame

Consider a hydrogen atom. Coulomb potential depends on the distance between the proton and electron only:

${\displaystyle V(r)={\frac {-Ze^{2}}{r}}}$

With this symmetry, the problem is reduced to that of a particle in a central potential:

${\displaystyle -{\frac {1}{2m}}\nabla ^{2}\psi ({\vec {r}})+{\frac {-Ze^{2}}{r}}\psi ({\vec {r}})=E\psi ({\vec {r}})}$

Using separation of variables, the solutions of the equation can be written into radial and angular parts:

${\displaystyle \Phi (r,\theta ,\phi )=R_{nl}(r)Y_{lm}(\theta ,\phi )}$

where ${\displaystyle l,}$ ${\displaystyle m}$, and ${\displaystyle n}$ are the orbital angular momentum, magnetic, and energy quantum numbers, respectively.

Now consider the Schrödinger equation for the proton and the electron:

${\displaystyle {\frac {\partial }{\partial t}}\Psi (x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},t)=-iH\Psi (x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},t)}$

A change of variables to relational and centre-of-mass coordinates yields

${\displaystyle {\frac {\partial \Psi (x,y,z,X,Y,Z)}{\partial t}}=-i[-{\frac {1}{2M}}\nabla _{c.o.m.}^{2}-{\frac {1}{2\mu }}\nabla _{rel}^{2}+V(x,y,z)]\Psi }$

where ${\displaystyle M}$ is the total mass and ${\displaystyle \mu }$ is the reduced mass. A final change to spherical coordinates followed by a separation of variables will yield the equation for ${\displaystyle \Phi (r,\theta ,\phi )}$ from above.

However, if the change of variables done early is now to be reversed, centre-of-mass needs to be put back into the equation for ${\displaystyle \Phi (r,\theta ,\phi )}$:

${\displaystyle r={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}}}}$

${\displaystyle \theta =\tan ^{-1}\left({\frac {\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}{z_{1}-z_{2}}}\right)}$

${\displaystyle \phi =\tan ^{-1}\left({\frac {y_{1}-y_{2}}{x_{1}-x_{2}}}\right)}$

${\displaystyle X={\frac {m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}}}$

${\displaystyle Y={\frac {m_{1}y_{1}+m_{2}y_{2}}{m_{1}+m_{2}}}}$

${\displaystyle Z={\frac {m_{1}z_{1}+m_{2}z_{2}}{m_{1}+m_{2}}}}$

The importance of this result is that it shows the wavefunction for the compound system is entangled, contrary to what one would normally think in a classical stand point. More importantly, it shows the energy of the hydrogen atom is not only associated with the electron but also with the proton, and the total state is not decomposable into a state for the electron and one for the proton separately.[1]

Superselection rules, in short, are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was originally introduced to impose additional restriction to quantum theory beyond those of selection rules. As an example, superselection rules for electric charges disallow the preparation of a coherent superposition of different charge eigenstates.

As it turns out, the lack of a reference frame is mathematically equivalent to superselection rules. This is a powerful statement because superselection rules have long been thought to have axiomatic nature, and now its fundamental standing and even its necessity are questioned. Nevertheless, it has been shown that it is, in principle, always possible (though not always easy) to lift all superselection rules on a quantum system.

During a measurement, whenever the relations between the system and the reference frame used is inquired, there is inevitably a disturbance to both of them, which is known as measurement back action. As this process is repeated, it decreases the accuracy of the measurement outcomes, and such reduction of the usability of a reference frame is referred to as the degradation of a quantum reference frame.[4][5] A way to gauge the degradation of a reference frame is to quantify the longevity, namely, the number of measurements that can be made against the reference frame until certain error tolerance is exceeded.

For example, for a spin-${\displaystyle j}$ system, the maximum number of measurements that can be made before the error tolerance, ${\displaystyle \epsilon }$, is exceeded is given by ${\displaystyle n_{max}\simeq \epsilon j^{2}}$. So the longevity and the size of the reference frame are of quadratic relation in this particular case.[6]

In this spin-${\displaystyle j}$ system, the degradation is due to the loss of purity of the reference frame state. On the other hand, degradation can also be caused by misalignment of background reference. It has been shown, in such case, the longevity has a linear relation with the size of the reference frame.[4]