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--- probability_and_uncertainty_in_quantum_mechanics [2021/02/08 07:21] – created admin
+++ probability_and_uncertainty_in_quantum_mechanics [2022/09/06 18:17] (current) – [1.vii.3 The Uncertainty Principle] admin
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 Since the particle will always be found somewhere if measured, this quantity needs to equal $1$.
-You will not always be lucky enough to be give neatly normalized wavefunctions, so what should you do if you are given a wavefunction $\phi(x,t)$ such that $\int_{-\infty}^{+\infty} |\phi(x,t)|^2\,\mathrm{d}x \neq 1$?  What I recommend doing is to always normalize the wavefunction as the first step in any calculation.  To do this, define $\psi(x,t) = A\phi(x,t)$, where $A$ is a constant that you determine by imposing $\int_{-\infty}^{+\infty} |\psi(x,t)|^2\,\mathrm{d}x = 1$, and then work with $\psi(x,t)$ for the rest of the calculation.
+You will not always be lucky enough to be given neatly normalized wavefunctions, so what should you do if you are given a wavefunction $\phi(x,t)$ such that $\int_{-\infty}^{+\infty} |\phi(x,t)|^2\,\mathrm{d}x \neq 1$?  What I recommend doing is to always normalize the wavefunction as the first step in any calculation.  To do this, define $\psi(x,t) = A\phi(x,t)$, where $A$ is a constant that you determine by imposing $\int_{-\infty}^{+\infty} |\psi(x,t)|^2\,\mathrm{d}x = 1$, and then work with $\psi(x,t)$ for the rest of the calculation.
 Equivalently, the Born rule can be generalized to unnormalized wavefunctions as follows.
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 ====== 1.vii.3 The Uncertainty Principle ======
 In 1927, Werner Heisenberg proposed the //**uncertainty principle**// relating our uncertainty about the position of a particle to our uncertainty about momentum.  For now, $\Delta x$ denotes a quantitative measure of our uncertainty about position and $\Delta p$ denotes a quantitative measure of our uncertainty about momentum.
+For a one-dimensional system, the uncertainty principle states
+\[\boxed{\Delta x \Delta p \geq \frac{\hbar}{2}}.\]
+For a three-dimensional system, this applies in all three coordinate directions.
+\[\boxed{\Delta x\Delta p_x\geq \frac{\hbar}{2},\qquad \Delta y\Delta p_y\geq \frac{\hbar}{2},\qquad \Delta z\Delta p_z\geq \frac{\hbar}{2}}.\]
+There are several possible meanings for these equations:
+  - //Preparation uncertainty//: At any given time, $\Delta x$, is our uncertainty about what the outcome of a position measurement would be were we to make it and $\Delta p$ is our uncertainty about what the outcome of a momentum measurement would be were we to make it.  (We can only actually make one of these two measurements.)
+  - //Measurement-disturbance//: If we make a measurement of $x$ with accuracy $\Delta x$ then the momentum will be disturbed by an amount $\Delta p$ and vice versa.
+  - //Joint measurement accuracy//: It is not possible to measure both $x$ and $p$ precisely at the same time.  If we attempt to make a measurement that tells us both $x$ and $p$ then the accuracy of those measurements is limited by $\Delta x \Delta p \geq \hbar/2.$
+All three of these are true in quantum theory.  2 and 3 have to be handled carefully because a quantum particle does not necessarily //have// a momentum or position that can be disturbed, so how can we tell the accuracy of a measurement of a quantity that does not necessarily exist before the measurement.  2 and 3 are the subject of ongoing research and controversy, but it is possible to come up with definitions of "disturbance" and "accuracy" such that they are true.
+Preparation uncertainty is the version that is usually proved in undergraduate quantum mechanics textbooks and we will do so in section 2.  However, Heisenberg originally argued for 2.
+Heisenberg's argument is a semi-classical argument that does not pass muster in full-blown quantum mechanics, but it will give you an idea of where the principle came from.  He imagines observing the position of an electron by scattering light off it.  He treats the electron classically, as if it had a well defined position and momentum, and applies quantum principles to the light.  This setup is often called the //**Heisenberg Microscope**//.  I will give a toy version of the argument here that gets the order of magnitude right.  Deriving precisely $\hbar/2$ as the limit requires a more detailed argument, which you can find at [[http://spiff.rit.edu/classes/phys314/lectures/heis/heis.html|this link]].
+To observe an electron's position with accuracy $\Delta x$, we need to use light of wavelength $\lambda \sim \Delta x$ or smaller.  By the de Broglie relation, this corresponds to a photon of momentum $p_{\text{light}} = h/\lambda \sim h/\Delta x$.
+To see the electron, the light must scatter off the electron, so there will be Compton scattering.  The change in momentum of the electron will be of the same order of magnitude as the initial momentum of the photon.  Therefore,
+\[\Delta p \sim p_{\text{light}} \sim \frac{h}{\Delta x} \qquad\qquad \Rightarrow \qquad\qquad \Delta x \Delta p \sim \hbar.\]
+The Heisenberg uncertainty relations can be generalized to any pair of complementary observables, i.e., variables that are canonically conjugate in classical mechanics.  In particlular, energy and time are complementary, so
+\[\boxed{\Delta E \Delta t \geq \frac{\hbar}{2}.}\]
+The interpretation of the energy-time uncertainty relation is more difficult than that of position and momentum because time is not treated as an observable in quantum mechanics, i.e. it is just a parameter that appears in the equation of motion and not something we usually calculate probabilities for.  There are several different versions of energy-time uncertainty that have appeared in the literature.  One version says that if your uncertainty about $E$ at time $t$ is $\Delta E$ and $\Delta t$ is the time it takes for the average energy $\bar{E}$ to change by an amount $\Delta E$ then $\Delta E\Delta t \geq \hbar/2$.  For a proof of this see the textbook "Quantum Mechanics" by Messiah, Section VIII.13.
+{{:question-mark.png?direct&50|}}
+====== In Class Activity ======
+  - Consider the wavefunction $\psi(x) = A\sqrt{|x|}e^{-x^2}$, where $A$ is a constant chosen such that $\int_{-\infty}^{+\infty} |\psi(x)|^2 = 1$.
+    - Find $A$.
+    - Determine $p(-1<x<1)$.