Due to Heisenberg uncertainty relation $$(\Delta x)(\Delta p) \geq \frac{\hbar}{2}$$ there exist an uncertainty in measurement of displacement and momentum. The state reach minimum uncertainty $$(\Delta x)(\Delta p) = \frac{\hbar}{2}$$ resemble the classical world.

From the derivation of uncertainty relation the inequality become equality only if these two relation hold:$$\hat{X}|\psi \rangle =c \hat{P}|\psi \rangle$$ and $$\langle \psi|[\hat{X},\hat{P}]_{+} |\psi \rangle $$ follow from the constraint, project the ket into $x$ basis to solve the differential equation obtain: $$\psi(x)=\psi(0)e^{i\langle P \rangle x/\hbar} e^{icx^2/2\hbar}$$ Which means minimum uncertainty wave function must be Gaussian.

Coherent state wave function in coordinate basis is $$\psi_{\alpha} (x)=\left(\frac{m\omega}{\pi \hbar}\right)^{1/4}e^{\frac{i}{\hbar}\langle p \rangle_{\alpha}x}e^{-\frac{m\omega}{2\hbar}(x-\langle x \rangle_{\alpha})^2}$$ which clearly satisfy the functional condition.

My problem as follows, is coherent state the unique state that is the most classical quantum mechanical state? If yes, how to prove uniqueness. If not, how to prove the non-uniqueness and what are some examples? What are the applications of these examples.


No. Squeezed vacuum states are also minimum uncertainty states for $x$ and $p$. They are also Gaussian in $x$ and $p$, but with different width in $x$ and $p$, still so that $\Delta x\Delta p=\hbar/2$.

They are used in metrology. The LIGO interferometers are injected with squeezed states so that $\Delta x$ is reduced. See for instance the paper:

Aasi, J., Abadie, J., Abbott, B. et al. Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light. Nature Photon 7, 613–619 (2013).

Note that such states are not eigenstates of the harmonic oscillator Hamiltonian, and the shape of the probability density $\vert\Psi(x,t)\vert^2$ depends on $t$.


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