\section{\label{sec:level1}Introduction}

\textit{Introduction.---}Ever since the establishment of quantum mechanics a century ago, the collapse of the wave function has been its central topic much debated. Numerous interpretations and theories have been proposed, including the Copenhagen interpretation \cite{bohr1928quantum, neumann1955mathematical}, many-worlds interpretation \cite{RevModPhys.29.454}, decoherence theories \cite{adler2003decoherence}, de Broglie--Bohm theory \cite{PhysRev.85.166, PhysRev.85.180}, quantum Bayesianism \cite{PhysRevA.65.022305, fuchs2010qbism, RevModPhys.85.1693}, and many others \cite{RevModPhys.85.471}. It is intriguing that if there exists a theory that is fully compatible with continuous quantum mechanics yet is capable of describing the dynamics of wave function collapse. After all, during the measurement process, the atoms and molecules in the measuring devices and the probed quantum objects are both governed by the Schr\"{o}dinger equation, hence their interaction and dynamics during the collapse of the wave function should also in principle be governed by the Schr\"{o}dinger equation.

In this Letter, based on the recent work \cite{hu2022quantum} of ours, we are going to show, starting with pure state cases, how the continuous collapse of the wave function process can actually be described by the Schr\"{o}dinger equation with a stochastic time-dependent Hamiltonian. We then solve analytically for the state-dependent Hamiltonian responsible for measurements on an arbitrary $n$-level system and position measurements on an harmonic oscillator in the ground state, and propose a few experimental schemes to verify and utilize the conclusions. To address the uniqueness of the Hamiltonian, we discuss how one can find all possible Hamiltonians that may be responsible for the wave function collapse such that the true Hamiltonian must be one of them. We then discuss and show how the above formalism can be used also to describe the wave function collapse for mixed quantum states. Our results show that the wave function collapse could be fully describable by a modified Schr\"{o}dinger equation where the Hamiltonian is state-dependent, which gives a new approach to applying control to measured systems.

\textit{Wave function collapse and continuous quantum measurement.}---While the projective measurement is usually treated for convenience to be happened instantaneously, it is reasonable to expect that actual physical processes always take some finite time to complete \cite{minev2019catch, mandelstam1945uncertainty, margolus1998maximum, deffner2013energy}. If we continuously measure a quantum object such that the density matrix $\hat{\rho}(t)$ of the object change gradually rather than all in a sudden, then this constitutes a continuous quantum measurement \cite{caves1987quantum, diosi1988continuous, brun2002simple, korotkov1999continuous, jacobs2006straightforward, chantasri2013action}. This process, when written in the infinitesimal form, can be described by some measurement operator $\hat{\mathcal{M}}_{\delta t}$ such that the density matrix $\hat{\rho}(t)$ at later time $t+\delta t$ changes to 
\begin{equation} \label{eq1}
\hat{\rho}(t+\delta t)=\frac{\hat{\mathcal{U}}_{\delta t} \hat{\rho}(t) \hat{\mathcal{U}}_{\delta t}^{\dagger}}{\operatorname{Tr}{[\hat{\mathcal{U}}_{\delta t} \hat{\rho}(t) \hat{\mathcal{U}}_{\delta t}^{\dagger}]}}, \quad \hat{\mathcal{U}}_{\delta t} \equiv e^{-i \hat{H}_s(t) \delta t} \hat{\mathcal{M}}_{\delta t},\end{equation}
where $\hat{H}_s(t)$ denotes the Hamiltonian of the system, $\hat{\mathcal{U}}_{\delta t}$ denotes the total evolution contributed by both $\hat{H}_s(t)$ and the measurement effect, and $\operatorname{Tr}{[\hat{\mathcal{U}}_{\delta t} \hat{\rho}(t) \hat{\mathcal{U}}_{\delta t}^{\dagger}}]$ serves as a normalization factor. Now if we consider the limit where the continuous quantum measurement disturbs the system strongly and happens extremely fast, then the whole process essentially becomes the projective measurement \cite{caves1987quantum} and can therefore be regarded as the continuous collapse of the wave function. In this extreme case, the effect from the measurement dominates such that we can take $\hat{H}_s(t) \approx 0,$ which will be assumed to be the case for the rest of the paper. Eq.~(\ref{eq1}) can thereby be rewritten as $\hat{\rho}(t+\delta t)=\frac{\hat{\mathcal{M}}_{\delta t} \hat{\rho}(t) \hat{\mathcal{M}}_{\delta t}^{\dagger}}{\operatorname{Tr}{[\hat{\mathcal{M}}_{\delta t} \hat{\rho}(t) \hat{\mathcal{M}}_{\delta t}^{\dagger}]}}$. For a given $\mathcal{\hat{M}}_{\delta t}$, we can then calculate the evolution of $\hat{\rho}(t)$ and record its evolution history as quantum trajectories \cite{Carmichael1993, wiseman1996quantum, brun2002simple, Bassi_2005, daley2014quantum}.

\textit{Reconstruct the Hamiltonian from the quantum trajectories.}---For simplicity, let us first consider the case where $\hat{\rho}(0)$ is initially pure. By requiring that during the projective measurement process, the quantum state does not become more mixed (since information is being extracted from the system), we can see that $\hat{\rho}(t)$ will remain pure at any later time $t$. This indicates that the evolution is unitary and can therefore be described by the Schr\"{o}dinger equation, at least effectively or phenomenologically. As a result, the problem becomes that with a given history of quantum trajectory $\ket{\psi(t)}$, which could either be calculated theoretically from certain $\hat{\mathcal{M}}_{\delta t}$, or reconstructed experimentally from measurement records (\textit{e.g}. via quantum tomography \cite{murch2013observing, weber2014mapping, jullien2014quantum, PhysRevX.6.041052}), how one can find an effective measuring Hamiltonian $\hat{H}(t)$ acting on the system alone that is responsible for the process. This problem, which is equivalent to solving the Schr\"{o}dinger equation inverse problem, has been solved in our recent work \cite{hu2022quantum}. We showed that for a given continuous quantum trajectory $\ket{\psi(t)}$, one can reconstruct its corresponding Hamiltonian $\hat{H}(t)$ by
\begin{equation} \label{eq2}
	\hat{H}(t) = i (\ketz{\p_t \tilde{\psi}(t)}\braz{\tilde{\psi}({t})}-\ketz{\tilde{\psi}(t)}\braz{\p_t\tilde{\psi}(t)})+\dot{\phi}(t) \mathbbm{1},
\end{equation}
where $\phi(t)\equiv\int-i\left\langle\partial_{t} \psi(t)|\psi(t)\right\rangle d t \in \mathbb{R}$ and $\ketz{\tilde{\psi}(t)}\equiv e^{i \phi(t)}\ket{\psi(t)}$, such that $\hat{H}(t)\ket{\psi(t)}=i \ket{\p_t \psi(t)}$. Then the general procedures for finding the ``measuring Hamiltonian'' responsible for the continuous wave function collapse of pure states are as follows:
\begin{enumerate}
	\item Assuming the process to be time-continuous, calculate theoretically (or measure experimentally) how the quantum state $\ket{\psi(t)}$ changes infinitesimally to $\ket{\psi(t+\delta t)}$, for instance, via certain  measurement operator $\hat{\mathcal{M}}_{\delta t}$,
\begin{equation} \label{eq3}
	\ket{\psi(t+\delta t)}=\f{\hat{\mathcal{M}}_{\delta t}}{\sqrt{\bra{\psi(t)}\hat{\mathcal{M}}_{\delta t}^\dagger \hat{\mathcal{M}}_{\delta t} \ket{\psi(t)}}} \ket{\psi(t)}.
\end{equation}
where ${\sqrt{\bra{\psi(t)}\hat{\mathcal{M}}_{\delta t}^\dagger \hat{\mathcal{M}}_{\delta t} \ket{\psi(t)}}}$ serves merely as a normalization factor.
	\item Calculate $\ket{\p_t \psi(t)}$ using its limit definition (or approximate with experimental data),
	\begin{equation} \label{eq4}
		\ket{\p_t \psi(t)}=\lim_{\delta t\to0} \f{\ket{\psi(t+\delta t)}-\ket{\psi(t)}}{\delta t}.
	\end{equation}
	\item Make use of Eq.~(\ref{eq2}) to reconstruct the measuring Hamiltonian $\hat{H}(t)$.
\end{enumerate}

Note that the measuring Hamiltonian is expected to generally be time-dependent and stochastic, such that it is compatible with the random nature of quantum mechanics and the Born rule. More interestingly, it is also state-dependent, meaning it will generally be different for different state. Such time- and state-dependency is determined by nature from the real-time interaction between the classical probe and the quantum object. In the above calculation. the stochasticity of the Hamiltonian comes from the stochasticity of quantum trajectories $\ket{\psi(t)}$. Physically, the stochasticity in the Hamiltonian may come from various sources, such as the vacuum fluctuation \cite{murch2013observing}, or more predominantly, just from the enormous number of degrees of freedom in the interaction between classical measuring devices and quantum objects. This interpretation differs from that of objective collapse theories, which holds that the quantum mechanics is intrinsically stochastic, even for the evolution of an isolated system \cite{RevModPhys.85.471}. It also differs from that of all sorts of hidden variable theories, since our formalism assumes it is the measuring device itself that causes the random nature of quantum mechanics.

Below we will show a few examples demonstrating our formalism.

\textit{Example I -- Two level system.}---To begin with, let us first consider the wave function of a two level system initially being at $\ket{\psi(0)}=a(0)\ket{0}+b(0)\ket{1}$ where $|a(0)|^2+|b(0)|^2=1$. We take our measuring operator $\hat{\mathcal{M}}_{\delta t}$ to be \cite{jacobs2014quantum}
\begin{equation} \label{eq5}
	\hat{\mathcal{M}}_{\delta t}=(\delta t / 2 \pi \tau)^{1 / 4} \exp \left[-\frac{\delta t}{4 \tau}\left(r_t \mathbbm{1}-\hat{\sigma}_{z}\right)^{2}\right],
\end{equation}
the physical meaning of which is that during the short time interval $[t,t+\delta t]$, we are measuring the $z$ component of the qubit with the measuring strength $1/\tau$, and the instantaneous readout $r_t \in (-\infty, +\infty)$ \cite{PhysRevLett.60.1351} at time $t$, which is a random variable, follows the conditional probability distribution $P(r_t\,|\,\rho(t))=\operatorname{Tr}[\hat{\rho}(t) \hat{\mathcal{M}}_{\delta t}\hat{\mathcal{M}}_{\delta t}^\dagger]$. The Gaussian part of the above equation indicates that the measurement effect is weighted towards the two eigenvalues $+1$ and $-1$ of the Pauli matrix $\hat{\sigma}_z$, so that the outputs of sequential applications of $\hat{\mathcal{M}}_{\delta t}$ will at last conform with the empirical facts that quantum states are always found to be at their eigenstates after von Neumann measurements. Note that different measurement approaches may give rise to different choices of $\hat{\mathcal{M}}_{\delta t}$. Assuming the measurement is strong so that the Hamiltonian of the system itself can be ignored, one then may calculate how $\ket{\psi(0)}$ will evolve under sequential applications of $\hat{\mathcal{M}}_{\delta t}$, until it stabilizes around its eigenstate $\ket{0}$ or $\ket{1}$.

By applying the aforementioned formalism we developed (Eq.~(\ref{eq2})-(\ref{eq4})),  we obtain the measuring Hamiltonian 
\begin{equation} \label{eq6}
\begin{aligned}
	\hat{H}(t)&=\begin{pmatrix}
	0&ia(t) b^*(t) \f{r_t}{\tau}\\
	-ia^*(t) b(t) \f{r_t}{\tau}&0
\end{pmatrix}\\
&=-\frac{r_t}{2 \tau}\left(-y(t) \hat{\sigma}_{x}+x(t) \hat{\sigma}_{y}\right),
\end{aligned}
\end{equation}
where $x(t)=-\operatorname{Im}[a(t)b^*(t)]/2$ and $y(t)=\operatorname{Re}[a(t)b^*(t)]/2$ are the components of the Bloch vector $\vec{v}_\rho$ of the density matrix. It can be easily verified that the $\hat{\mathcal{M}}_{\delta t}$ introduced in Eq.\,(\ref{eq5}) and the Hamiltonian we just obtained result in exactly the same evolution of $\ket{\psi(t)}$. The random variable $r_t$ is chosen from the distribution $P(r_t\,|\,\rho(t))$ depending on $a(t)$ and $b(t)$, both of which go into $H(t)$, which in turn determines the value of $a(t+\delta t)$ and $b(t+\delta t)$.

\begin{figure}[!htp]
\subfloat{\includegraphics[width=.36\textwidth,scale=1]{plot01.eps}\hfill}
\caption{\label{fig:epsart1} The Bloch vector $\vec{v}_\rho$ of qubit is always orthogonal to and rotates around the Bloch vector $\vec{v}_H$ of the Hamiltonian given by Eq.\,(\ref{eq6}) during the measurement process, as visualized for two-level case. This is also the case in $n$-level system. The qubit then moves stochastically on the displayed longitudinal arc, until it stabilizes at $\ket{0}$ or $\ket{1}$. When one simultaneously measure the qubit along different axes (e.g. $\hat{x}$ and $\hat{z}$), it will make the qubit subject to the rotations of two different axes, $\vec{v}_{H_1}$ and $\vec{v}_{H_2}$, which is highly chaotic, and also makes the original eigenstates no longer be stabilization points.}
\end{figure}

There are a few comments worth mentioning regarding the Hamiltonian. First, it has instantaneous eigenvalues $E_{\pm}=\pm|a(t)||b(t)| \frac{r_t}{\tau}$ and instantaneous energy variance $[\Delta H(t)]^2= |a(t)|^2 |b(t)|^2 \f{r_t^2}{\tau^2}$ such that when we make a very strong measurement (\textit{i.e.} $1/\tau \to \infty$), its eigenvalues and energy variance diverge (or more physically, become very large). Not only does this validate our earlier treatment $H_s(t) \approx 0$, but also it is consistent with the Heisenberg uncertainty principle $\Delta H \Delta t \gtrsim \hbar/2$, and the idea of quantum speed limit \cite{mandelstam1945uncertainty, margolus1998maximum, deffner2013energy, hu2022quantum}, that a rapidly changing quantum state inevitably has a large energy variance. The fact that $[\Delta H(t)]^2 \to \infty$ as $1/\tau \to \infty$ also echoes our earlier claim that all physical processes take finite amount of time to happen, since in reality the energy supplied during measurement cannot be infinite. 

Second, when the quantum state reaches one of its eigenstate, $\ket{0}$ or $\ket{1}$, which means $a(t)=0$ or $b(t)=0$, the measuring Hamiltonian vanishes, so that the quantum state is stabilized and no longer evolves, had we assumed $H_s(t)\approx 0$. In addition, while the mean energy of the Hamiltonian is zero, an arbitrary real function $\dot{\phi}(t)$ can be added to the diagonal term such that $H(t) \to H(t)+\dot{\phi}(t) \mathbbm{1}$, which only changes the global $U(1)$ phase of $\ket{\psi(t)}$ and therefore can also be used to describe the measuring process. Finally, if we plot on the Bloch sphere the evolution of the qubit during the measurement, we notice that the qubit rotates exactly around the axis $-y(t)\hat{x}+x(t)\hat{y}$ pointed by the Bloch vector $\vec{v}_H$ of the Hamiltonian, and that the Bloch vector $\vec{v}_\rho=x(t)\hat{x}+y(t)\hat{y}$ of the qubit is orthogonal to $\vec{v}_H$ (Fig.\,\ref{fig:epsart1}). It can be shown that this conclusion actually also applies to the measurement on $n$-level system, if we reconstruct the Hamiltonian via Eq.\,(\ref{eq2}) (Supplemental Materials I). Moreover, this observation illustrates why we cannot make precise simultaneous measurements in the different bases, such as $\hat{x}$ and $\hat{z}$, as this will force the qubit to stochastically rotate around ``two'' different axes, $\vec{v}_{H_1}$ and $\vec{v}_{H_2}$, making the total evolution highly chaotic. It also makes the measuring Hamiltonian no longer vanish at the eigenstates of $\hat{\sigma}_x$ and $\hat{\sigma}_z$, so the qubit will continue to evolve and escape from its original eigenstates, making the readouts spread over a large region \cite{ruskov2010qubit, chantasri2018simultaneous}.

To experimentally verify and utilize the conclusions drawn from the above discussions, one can implement real-time quantum feedback control to ``freeze'' the qubit under the continuous quantum measurement. For instance, for a given known initial state $\ket{\psi(0)}$, one can apply a real-time feedback control Hamiltonian $\hat{H}_c(t)=-(\hat{H}_s(t)+\hat{H}_m(t))$, where $\hat{H}_m(t)$ denotes the measuring Hamiltonian defined by Eq.\,(\ref{eq6}). The total Hamiltonian acting on the qubit during the measurement then is $\hat{H}_s(t)+\hat{H}_m(t)+\hat{H}_c(t)=0$, meaning the qubit will be ``frozen''. Experimentally such scheme can be more easily implemented via kicked quantum nondemolition measurements, e.g., by applying periodic weak measurements with certain time interval $T$ such that $\hat{U}(T)=\exp(-i \hat{H}_s T)=I$, where we have assumed $\hat{H}_s$ to be time-independent. In this way, the contribution of the evolution from $\hat{H}_s$ is effectively zero in each measurement snapshot, and 
the waiting time between two successive measurements provides sufficient room for feedback controls to be implemented \cite{PhysRevB.74.085307}.

Another way to experimentally verify and utilize the conclusions is related to the most probable path. In \cite{PhysRevA.88.042110}, it is shown by using the stochastic path integral and the variational principle, and later verified experimentally in \cite{weber2014mapping}, that there exists a most probable path of a continuously monitored quantum system, the evolution of which are subject to the effects from both the Hamiltonian of system and the measuring processes. We can thereby solve for an effective Hamiltonian $\hat{H}_\text{eff}(t)$ responsible for such most probable path via Eq.\,(\ref{eq2}-\ref{eq4}), and by applying the counter-Hamiltonian $-\hat{H}_\text{eff}(t)$, to cancel out the most probable path. Experiments can then be designed to check whether the most probable path vanishes under the application of $-\hat{H}_\text{eff}(t)$ so as to verify if $\hat{H}_\text{eff}(t)$ describes well the dynamics of a continuously monitored quantum system. In Supplemental Materials \rom{2}, we provide an example to show the calculation of $\hat{H}_\text{eff}(t)$ of a continuously monitored qubit with its Hamiltonian $\hat{H}=\f{\epsilon}{2}\hat{\sigma}_z$.

\textit{Example II -- $n$-level system.}---Now let us consider the measurement on an arbitrary $n$-level system initially at $\ket{\psi(0)}=\sum_i^n a_i(0)\ket{i}$ where $\sum_i^n|a_i(0)|^2=1$. Similarly, let the measurement operator be $\hat{\mathcal{M}}_{\delta t}=\sum_{i}^{n} M_{i}(t)|i\rangle\langle i|$ where $
	M_{i}(t)=(\delta t / 2 \pi \tau)^{1 / 4} \exp [-\frac{\delta t}{4 \tau}\left(r_t-\lambda_{i}\right)^{2}]$. The $\lambda_i$ represents the $i$-th eigenvalue corresponding to the $i$-th eigenstate of all possible outcomes. By using again Eq.\,(\ref{eq2})-(\ref{eq4}), it can be shown (Supplemental Materials \rom{3}) that the measuring Hamiltonian responsible for the wave function collapse is given by (do not confuse the imaginary number $i$ with the summation index $i$)
	\begin{equation}\label{eq7}
		[H(t)]_{i j}=i \frac{a_{i}(t) a^*_{j}(t)}{4 \tau} \sum_{k}\left|a_{k}(t)\right|^{2}\left(\eta_{i k}+\eta_{k j}\right),
	\end{equation}
	where $\eta_{i j} \equiv\left(\lambda_{i}-\lambda_{j}\right)\left(2 r_t-\lambda_{i}-\lambda_{j}\right)$. The instantaneous energy variance $[\Delta H(t)]^2$ is calculated to be
\begin{equation} \label{eq8}
	[\Delta H(t)]^2=\sum_{j}^{n}\left[\frac{\left|a_{j}(t)\right|^{2}}{16 \tau^{2}}\left(\sum_{i}^{n}\left|a_{i}(t)\right|^{2} \eta_{ij}\right)^{2}\right].
\end{equation}


\textit{Example III -- harmonic oscillator at the ground state.}---Let us consider the ground state wave function of an harmonic oscillator peaked around $x_0$ and at time $t=0$,
\begin{equation}
	\psi(x, 0)=\left(\frac{m \omega}{\pi \hbar}\right)^{1 / 4} \exp \left(-\frac{m \omega}{2 \hbar}\left(x-x_{0}\right)^{2}\right).
\end{equation}
The measurement operator for a weak position measurement can be given by \cite{caves1987quantum} $\hat{\mathcal{M}}_{\delta t}=\int(\delta t / 2 \pi \tau)^{1 / 4} \exp \left[-\frac{\delta t}{4 \tau}(r_t-x)^{2}\right]|x\rangle\langle x| d x$. Expressing the wave function $\psi(x, 0)$ by the ket state $\ket{\psi(0)}=\int \psi(x,0)\ket{x}dx$, and applying Eq.\,(\ref{eq2})-(\ref{eq4}), we obtain (Supplemental Materials \rom{4}) the measuring Hamiltonian at time $t=0$,
\begin{widetext}
\begin{equation}\label{eq10}
	\hat{H}(0)=i\hbar\int \f{\eta_{x x^\prime}}{4\tau}\sqrt{\f{m \omega}{\pi \hbar}}\exp{\left( -\f{m \omega ((x-x_0)^2+{(x^\prime}-x_0)^2)}{2\hbar}\right)} \ket{x}\bra{x^\prime}dxdx^\prime,
\end{equation}
\end{widetext}
where $\eta_{xx^\prime} \equiv (x-x^\prime)(2r_t-x-x^\prime)$, which has instantaneous energy uncertainty 
\begin{equation}\label{eq11}
	\Delta H(0)=\sqrt{\frac{\hbar \left(\hbar+4 m \omega (r_t-x_0)^{2} \right)}{32 m^{2} \omega^{2}\tau^{2} }} \hbar.
\end{equation}
It can be shown that at time $t=\delta t$, the wave function is still a Gaussian, so we can apply the same procedure repeatedly, which in turn indicates that the wave function will always be a Gaussian at any later time. The corresponding measuring Hamiltonian $\hat{H}(t)$ will therefore also share a similar form as $\hat{H}(0)$, except that the $x_0$ and $\omega$ in $\hat{H}(0)$ will need to be replaced by some other numbers. More specifically, it can be shown by solving a differential equation that the position uncertainty $\Delta X(t)$ at time $t$ can be related to $\Delta X(0)$ by $[\Delta X(t)]^{2}=\frac{[\Delta X(0)]^{2} \tau}{[\Delta X(0)]^{2} t+\tau}$. By replacing $\omega$ with $\Delta X(t)$ and using Eq.\,(\ref{eq11}), we obtain
\begin{equation} \label{eq12}
	 \Delta H(t)=\frac{\sqrt{2 (r_t-x_t)^{2} [\Delta X(t)]^{2}+[\Delta X(t)]^{4}}}{2 \sqrt{2} \tau} \hbar,
	 \end{equation}
which approximates to $|r_t-x_t|\hbar/(2\sqrt{rt})$ as $\tau \to 0.$
Such expression of $\Delta H(t)$ gives the energetic resources needed to quantify a given measurement dynamics. It is interesting to see that if the readout $r_t$ is too far away from the mean position $x_t$, then $\Delta H(t)$ becomes large. It is also worth noting that the Hamiltonian $\hat{H}(0)$ is non-local, since $\bra{x}\hat{H}(0)\ket{x^\prime}$ is generally nonzero for $x \neq x^\prime$. This seems to imply that position measurements are intrinsically non-local in quantum mechanics --- imagine the extreme case that the position measurement on a free particle in a very large room (e.g. $\sim$100 km$^2$) and the particle is found almost instantaneously to be somewhere in the room, the process of which would require detectors placed all over the room with good timing coordination such that the measuring Hamiltonian has to be regarded effectively non-local.

\textit{Discussions on the non-uniqueness of the measuring Hamiltonian.}---The measuring Hamiltonian may not be unique. Two totally different Hamiltonians with different spectra may result in exactly the same quantum evolution. For example, both $\hat{H}=-\hat{\sigma}_z$ and $\hat{H}(t)=\f{1}{3}\mathbbm{1}-\f{2}{9}\sqrt{2}\cos{(2t)}\hat{\sigma}_x+\f{2}{9}\sqrt{2}\sin{(2t)}\hat{\sigma}_y-\f{8}{9}\hat{\sigma}_z$ drive $\ket{\psi(t)}=\f{e^{it}}{\sqrt{3}}\ket{0}+\sqrt{\f{2}{3}}e^{-i t}\ket{1}$, and there is no way to tell which one is the actual Hamiltonian responsible for the driving just from (hypothetically) observing the quantum trajectory of $\ket{\psi(t)}$. This implies that the measuring Hamiltonian we obtained from Eq.~(\ref{eq2})-(\ref{eq4}) may not be the actual Hamiltonian responsible for the wave function collapse. Is there a way to find the true measuring Hamiltonian? With only the information from the quantum trajectories $\ket{\psi(t)}$, the answer is negative. Nevertheless, we prove the following theorem in the Supplemental Materials \rom{5}.

\textit{Theorem}. \textit{Let $\ket{\psi(t)}\neq0$ be an arbitrary pure quantum state at time $t$. Let $V$ be the linear space formed by all Hermitian operators $\hat{T}(t)$ satisfying $\hat{T}(t)\ket{\psi(t)}=0.$ If one can find at least one Hamiltonian $\hat{H}(t)$ satisfying $\hat{H}(t)\ket{\psi(t)}=i \ket{\p_t \psi(t)}$, then one can find all possible Hamiltonians $\hat{H}^\prime(t)$ satisfying $\hat{H}^\prime(t)\ket{\psi(t)}=i \ket{\p_t \psi(t)}$. In particular, $\hat{H}^\prime(t)$ can be given by $\hat{H}^\prime(t)=\hat{H}(t)+\hat{T}(t)$ where $\hat{T}(t)$ is arbitrarily chosen from $V$.}

The above theorem implies that even though we may never be able to know the actual measuring Hamiltonian, we can be assured that it must be one of the possible $\hat{H}^\prime(t)$, if we are able to solve for all $\hat{T}(t)$. Moreover, all such Hamiltonians $H^\prime(t)$ should have the same energy variance equal to $\braket{\p_t \psi(t)}{\p_t \psi(t)}-|\braket{\psi(t)}{\p_t\psi(t)}|^2$.

\textit{Measurement on a mixed quantum state.}---Here we briefly discuss the generalization of the measurement on a mixed quantum state. Consider an ensemble of (proper) mixed quantum states $\rho(0)=\sum_i^n p_i(0) \ket{\psi_i(0)}\bra{\psi_i(0)}$ at $t=0$, which is prepared by mixing up $n$ orthonormal pure states $\ket{\psi_i(0)}$ with relative population $p_i(0)$, such that for a quantum object randomly selected from the ensemble, there is a classical probability $p_i(0)$ for the selected object to be in $\ket{\psi_i(0)}$. Now suppose one makes a measurement on the selected object, then according to the earlier theory we developed, one concludes that there should be a Hamiltonian responsible for the measuring process, since the selected quantum object is actually pure --- it is just that one has lost track (information) of the state. In this case, one can describe the measuring process by telling that at time $t=0$, there is a probability $p_i(0)$ that the measuring Hamiltonian is given by $\hat{H}_{\ket{\psi_i(0)}}$. Such description $\{ p_i(0)\}$ needs to be Bayesian updated at later time $t$ to $\{ p_i(t)\}$ once a series of readouts $\{r_t\}$ are accumulated, indicating the process of purification. Note that in this scheme, the selected quantum object actually remains pure all the time, corresponding to a unitary process. It is the observer's updating knowledge of the state that makes the evolution appear non-unitary for the observer.

\textit{Conclusion.}---In this Letter, we demonstrate how one can describe the wave function collapse process by using the Schr\"{o}dinger equation and the formalism we developed in \cite{hu2022quantum} and Eq.~(\ref{eq2})-(\ref{eq4}). We solve analytically the measuring Hamiltonian (Eq.(\ref{eq6}),(\ref{eq7}), and (\ref{eq10})) responsible for the process in varied cases, and calculate its instantaneous energy variance (Eq.(\ref{eq8}) and (\ref{eq12})). We then discuss the non-uniqueness of the measuring Hamiltonian, and how our results can also be applied to describe the wave function collapse process for mixed quantum states.

\textit{Acknowledgement.}---We are grateful to the support from the Army Research Office (ARO) under the grant W911NF-22-1-0258.

