[Physics] Is it possible to reconstruct the Hamiltonian from knowledge of its ground state wave function

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Is it possible to "construct" the Hamiltonian of a system if its ground state wave function (or functional) is known? I understand one should not expect this to be generically true since the Hamiltonian contains more information (the full spectrum) than a single state vector. But are there any special cases where it's possible to obtain the Hamiltonian? Some examples would be really helpful.

Best Answer

IF you know that your Hamiltonian is of the form $$ \hat H=\frac{-\hbar^2}{2m}\nabla^2+V(\mathbf r) \tag 1 $$ for a single massive, spinless particle, then yes, you can reconstruct the potential and from it the Hamiltonian, up to a few constants, given any eigenstate. To be more specific, the ground state $\Psi_0(\mathbf r)$ obeys $$ \hat H\Psi_0(\mathbf r) =\frac{-\hbar^2}{2m}\nabla^2\Psi_0(\mathbf r)+V(\mathbf r)\Psi_0(\mathbf r) =E_0 \Psi_0(\mathbf r), $$ which means that if you know $\Psi_0(\mathbf r)$ then you can calculate its Laplacian to get $$ \frac{ \nabla^2 \Psi_0(\mathbf r) }{ \Psi_0(\mathbf r) } = \frac{2m}{\hbar^2}\left(V(\mathbf r)-E_0\right). $$ If you know the particle's mass, then you can recover $V(\mathbf r)-E_0$, and this is all you really need (since adding a constant to the Hamiltonian does not change the physics).

However, it's important to note that this procedure guarantees that your initial $\Psi_0$ will be an eigenstate of the resulting hamiltonian, but it does not preclude the possibility that $\hat H$ will admit a separate ground state with lower energy. As a very clear example of that, if $\Psi_0$ is a 1D function with a node, then (because 1D ground states have no nodes) you are guaranteed a unique $V(x)$ such that $\Psi_0$ is an eigenstate, but it will never be the ground state.

If you don't know that your Hamiltonian has that structure, there is (in the general case) no information at all that you can extract about the Hamiltonian from just the ground state.

  • As a simple example, without staying too far from our initial Hamiltonian in $(1)$, consider that Hamiltonian in polar coordinates, $$\hat H=\frac{-\hbar^2}{2m}\left(\frac{1}{r^2}\frac{\partial}{\partial r} r^2\frac{\partial}{\partial r} + \frac{1}{\hbar ^2r^2}L^2\right)+V(r),$$ where I'm assuming $V(\mathbf r)=V(r)$ is spherically symmetric, and encapsulating the angular dependence into the total angular momentum operator $L^2$.

    Suppose, then, that I give you its ground state, and that it is an eigenstate of $L^2$ with eigenvalue zero (like e.g. the ground state of the hydrogenic Hamiltonian). How do you tell if the Hamiltonian that created it is $H$ or a similar version, $$\hat H{}'=\frac{-\hbar^2}{2m}\frac{1}{r^2}\frac{\partial}{\partial r} r^2\frac{\partial}{\partial r} +V(r),$$ with no angular momentum component? Both versions will have $\Psi_0$ as a ground state (though here $\hat H'$ will have a wild degeneracy on every eigenspace, to be fair). Carrying on with this thought, what about $$\hat H{}''=\frac{-\hbar^2}{2m}\left(\frac{1}{r^2}\frac{\partial}{\partial r} r^2\frac{\partial}{\partial r} + f(r)L^2\right)+V(r),$$ where I've introduced an arbitrary real function $f(r)$ behind the angular momentum? This won't affect the $\ell=0$ states, but it will take the rest of the spectrum to who knows where. (In fact, you can even tack on an arbitrary function of $L_x$, $L_y$ and $L_z$, while you're at it.)

  • A bit more generally, any self-adjoint operator which vanishes on $\left|\Psi_0\right>$ can be added to the Hamiltonian to get you an operator that has $\left|\Psi_0\right>$ as an eigenstate. As a simple construction, given any self-adjoint operator $\hat A$, the combination $$\hat H {}''' = E_0 \left|\Psi_0\right>\left<\Psi_0\right| + \left(\mathbf 1 - \left|\Psi_0\right>\left<\Psi_0\right| \right) \hat A \left(\mathbf 1 - \left|\Psi_0\right>\left<\Psi_0\right| \right) $$ (where the factors in brackets are there to modify $\hat A$ into vanishing at $\left|\Psi_0\right>$ and its conjugate) will always have $\left|\Psi_0\right>$ as an eigenstate.

Even if you know all the eigenstates, it's still not enough information to reconstruct the Hamiltonian, because they do not allow you to distinguish between, say, $\hat H$ and $\hat{H}{}^2$. On the other hand, if you know all the eigenstates and their eigenvalues, then you can simply use the spectral decomposition to reconstruct the Hamiltonian.

In general, if you really insist, there is probably a trade-off between what you know about the Hamiltonian's structure (e.g. "of the form $\nabla^2+V$" versus no information at all) and how many of the eigenstates and eigenvalues you need to fully reconstruct it (a single pair versus the whole thing), particularly if you allow for approximate reconstructions. Depending on where you put one slider, you'll get a different reading on the other one.

However, unless you have a specific problem to solve (like reconstructing a Hamiltonian of vaguely known form from a specific set of finite experimental data) then it's definitely not worth it to explore the details of this continuum of trade-offs beyond the knowledge that it exists and the extremes I noted above.