Usually Heisenberg uncertainty principle is formulated in terms of statistics of measurement results when the measurements are independent and performed on identically prepared quantum systems. For position and momentum measurement the Heisenberg uncertainty principle leads to the inequality \[ \Delta x\Delta p\geq\frac{\hbar}{2}\,, \] where \(\Delta x\) and \(\Delta p\) are standard deviations of the measured values. Here we will consider a measurement process where we try to measure the position and momentum of a single quantum system at the same time. To estimate the measurement errors we will describe the measurement similarly as in a simple von Neumann measurement.
Let us consider the interaction of the quantum system with the microscopic part of the detector described by the Hamiltonian \[ H=\gamma_1 p_1 x+\gamma_2 p_2 p\,, \] where \(x\) and \(p\) are the position and momentum of the quantum system, \(p_i\) are the detector momenta conjugate to the detector coordinates \(q_i\). Large parameters \(\gamma_1\) and \(\gamma_2\) describe the strength of the interaction. Before the measurement the detector is in a separable pure state with the wave function \(\phi_1(q_1)\phi_2(q_2)\) that has well defined coordinates \(q_i\). If the wave function of the quantum system before the measurement is \(\psi(x)\) then the initial state of system and the detector is \[ \Psi(x, q_1, q_2)=\psi(x)\phi_1(q_1)\phi_2(q_2)\,. \] Assuming that the duration of the interaction is \(\tau\), after the interaction the state becomes \[ \Psi'(x, q_1, q_2)=\exp\left(-\frac{\mathrm{i}}{\hbar}\gamma_1\tau p_1 x-\frac{\mathrm{i}}{\hbar}\gamma_2\tau p_2 p\right) \psi(x)\phi_1(q_1)\phi_2(q_2)\,. \] According to Baker-Campbell-Hausdorff (BCH) formula, when the commutator \([A,B]\) commutes with \(A\) and \(B\), we have the equality \[ \mathrm{e}^{A+B}=\mathrm{e}^{A}\mathrm{e}^{B}\mathrm{e}^{-\frac{1}{2}[A,B]}\,. \] Taking int account the commutator \([x,p]=\mathrm{i}\hbar\) from BCH formula we obtain \[ \Psi'(x, q_1, q_2)=\exp\left(-\frac{\mathrm{i}}{\hbar}\gamma_1\tau p_1 x\right) \exp\left(-\frac{\mathrm{i}}{\hbar}\gamma_2\tau p_2 p\right) \exp\left(\frac{i}{2\hbar}\gamma_1\gamma_2\tau^2 p_1 p_2 \right)\psi(x)\phi_1(q_1)\phi_2(q_2)\,. \] Introducing the wave function of the detector \[ \Phi(q_1, q_2; \xi)=\exp\left(\frac{2i}{\hbar^2}\xi p_1 p_2 \right)\phi_1(q_1)\phi_2(q_2) \] we can write \[ \Psi'(x, q_1, q_2)=\exp\left(-\frac{\mathrm{i}}{\hbar}\gamma_1\tau p_1 x\right) \exp\left(-\frac{\mathrm{i}}{\hbar}\gamma_2\tau p_2 p\right)\psi(x)\Phi(q_1, q_2; \xi)\,. \] Here \[ \xi=\frac{\hbar}{4}\gamma_1\gamma_2\tau^2\,. \] Going into momentum representation for the measured quantum system \[ \tilde{\psi}(p)=\frac{1}{\sqrt{2\pi\hbar}}\int\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}px}\psi(x)\,\mathrm{d}x \] we get the state of the quantum system and the detector after the interaction \[ \Psi'(x, q_1, q_2)=\frac{1}{\sqrt{2\pi\hbar}}\int\mathrm{e}^{\frac{i}{\hbar}px}\tilde{\psi}(p) \Phi(q_1 - \gamma_1\tau x,q_2 - \gamma_2\tau p; \xi)\,\mathrm{d}p\,. \]
Applying the projective measurement of the detector postions \(q_1\) and \(q_2\) by the rest of the detector with the results \(q_1^{\prime}\) and \(q_2^{\prime}\) yields the post-measurement wave fuction of the quantum system \[ \psi(x; q_1^{\prime},q_2^{\prime})=\frac{1}{\sqrt{P(q_1^{\prime}, q_2^{\prime})}} \Psi'(x, q_1^{\prime}, q_2^{\prime})\,, \] where \[ P(q_1^{\prime}, q_2^{\prime})=\int|\Psi'(x, q_1^{\prime},q_2^{\prime})|^2\,\mathrm{d}x \] is the probability to obtain outcomes \(q_1^{\prime}\) and \(q_2^{\prime}\). If the initial wave functions of the detector \(\phi_i\) are narrowly centered around \(q_{i,0}\), then the measurement results are \begin{align*} x' & =\frac{1}{\gamma_1\tau}(q_1^{\prime} - q_{1,0})\,,\\ p' & =\frac{1}{\gamma_2\tau}(q_2^{\prime} - q_{2,0})\,. \end{align*} The probability distribution of measurement results is \(P(x',p')=\gamma_1\gamma_2\tau^2 P(q_1^{\prime}, q_2^{\prime})\).
Let us calculate the function \(\Phi\) for a particular case, Gaussian initial wave functions of the detector \[ \phi_i(q_i)=\frac{1}{(2\pi)^{\frac{1}{4}}\sqrt{\epsilon_i}} \exp\left(-\frac{1}{4\epsilon_i^2}(q_i - q_{i,0})^2\right)\,. \] The momentum representation of these wave functions is \[ \tilde{\phi}_i(p_i)=\left(\frac{2}{\pi}\right)^{\frac{1}{4}}\sqrt{\frac{\epsilon_i}{\hbar}} \exp\left(-\frac{\epsilon_i^2}{\hbar^2}p_i^2 - \frac{\mathrm{i}}{\hbar}p_i q_{i,0}\right)\,. \] Using the momentum representation of the detector wave functions \(\phi_i\), the function \(\Phi\) can be expressed as \[ \Phi(q_1, q_2; \xi)=\frac{1}{2\pi\hbar}\int\mathrm{d}p_1\int\mathrm{d}p_2\,\mathrm{e}^{\frac{\mathrm{i}}{\hbar}p_1 q_1 +\frac{\mathrm{i}}{\hbar}p_2 q_2 +\frac{2\mathrm{i}}{\hbar^2}\xi p_1 p_2}\tilde{\phi}_1(p_1)\tilde{\phi}_2(p_2)\,. \] We obtain \[ \Phi(q_1, q_2; \xi)=\frac{1}{\sqrt{2\pi\epsilon_1^{\prime}\epsilon_2^{\prime}}} \exp\left(-\frac{(q_1 - q_{1,0})^2}{4\epsilon_1^{\prime 2}}-\frac{(q_2 - q_{2,0})^2}{4\epsilon_2^{\prime 2}} -\mathrm{i}\xi\frac{(q_1 - q_{1,0})(q_2 - q_{2,0})}{2\epsilon_1\epsilon_2\epsilon_1^{\prime}\epsilon_2^{\prime}}\right)\,, \] where \[ \epsilon_i^{\prime}=\epsilon_i \sqrt{1+\frac{\xi^2}{\epsilon_1^2\epsilon_2^2}}\,. \] If we perform only the position measurement or only the momentum measurement, the measurement errors would be \[ \Delta x_0=\frac{\epsilon_1}{\gamma_1\tau}\,,\qquad\Delta p_0=\frac{\epsilon_2}{\gamma_2\tau}\,. \] From the expression for the function \(\Phi\) we see that trying to measure both position and momentum simultaneously the measurement errors are \[ \Delta x=\frac{\epsilon_1^{\prime}}{\gamma_1\tau}\,,\qquad\Delta p=\frac{\epsilon_2^{\prime}}{\gamma_2\tau}\,. \] The product of measurement errors is \[ \Delta x\Delta p=\Delta x_0\Delta p_0\left(1+\left(\frac{\hbar}{4}\frac{1}{\Delta x_0\Delta p_0}\right)^2\right)\,. \] The minimum of \(\Delta x\Delta p\) is at \[ \Delta x_0\Delta p_0=\frac{\hbar}{4} \] and takes the value \[ \Delta x\Delta p=\frac{\hbar}{2}\,. \] Thus for all parameters \[ \Delta x\Delta p\geq\frac{\hbar}{2}\,. \] That is, the product of measurement errors is has a lower limit.
We have the commutator \[ \left[q_1,\exp\left(\frac{2\mathrm{i}}{\hbar^2}\xi p_1 p_2\right)\right] =\mathrm{i}\hbar\frac{\partial}{\partial p_1}\exp\left(\frac{2\mathrm{i}}{\hbar^2}\xi p_1 p_2\right)= -\frac{2}{\hbar}\xi p_2\exp\left(\frac{2\mathrm{i}}{\hbar^2}\xi p_1 p_2\right)\,. \] Using this expression we obtain \[ \langle\Phi|q_1|\Phi\rangle=\langle\phi_1|q_1|\phi_1\rangle-\frac{2}{\hbar}\xi\langle\phi_2|p_2|\phi_2\rangle \] and \[ \langle\Phi|q_1^2|\Phi\rangle=\langle\phi_1|q_1^2|\phi_1\rangle -\frac{4}{\hbar}\xi\langle\phi_1|q_1|\phi_1\rangle\langle\phi_2|p_2|\phi_2\rangle +\frac{4}{\hbar^2}\xi^2\langle\phi_2|p_2^2|\phi_2\rangle\,. \] The dispersion of \(q_1\) is \[ \Delta q_1^2=\langle\Phi|q_1^2|\Phi\rangle-\langle\Phi|q_1|\Phi\rangle^2=\delta q_1^2+\frac{4}{\hbar^2}\xi^2\delta p_2^2\,, \] where \(\delta a_i^2=\langle\phi_i|a_i^2|\phi_i\rangle-\langle\phi_i|a_i|\phi_i\rangle^2\) is the initial dispersion of observable \(a_i\). Multiplying the dispersions we get \[ \Delta q_1^2\Delta q_2^2=\delta q_1^2\delta q_2^2+\frac{4}{\hbar^2}\xi^2\delta q_1^2\delta p_1^2 +\frac{4}{\hbar^2}\xi^2\delta q_2^2\delta p_2^2+\frac{16}{\hbar^4}\xi^4\delta p_1^2\delta p_2^2\,. \] From the uncertainty principle it follows that \[ \delta q_1^2\delta p_1^2\geq\frac{\hbar^2}{4}\,. \] Applying this inequality we obtain \[ \Delta q_1^2\Delta q_2^2\geq2\xi^2+\delta q_1^2\delta q_2^2+\frac{\xi^4}{\delta q_1^2\delta q_2^2}\,. \] The last two terms have a minimum when \(\delta q_1^2\delta q_2^2=\xi^2\), threfore \[ \Delta q_1^2\Delta q_2^2\geq 4\xi^2\,. \]