Differences

This shows you the differences between two versions of the page.

--- wave-particle_duality [2021/02/08 20:25] – [1.vi.2 The Double Slit Experiment] admin
+++ wave-particle_duality [2022/09/06 18:08] (current) – [Quantum Particles] admin
@@ Line 83: / Line 83: @@
 We observe a wave-like interference between the two possible trajectories //only// if we cannot tell which trajectory was actually taken.  Otherwise, we observe particle like properties, with no interference patter.
-Niels Bohr introduced the principle of //**complementarity**/// to describe this:  We need both particle and wave concepts to describe quantum systems, but we can only ever observe one of them at a time, depending on the experimental arrangement.  It is not a case of either/or or both/and.  They are complementary aspects of the same physical system.
+Niels Bohr introduced the principle of //**complementarity**// to describe this:  We need both particle and wave concepts to describe quantum systems, but we can only ever observe one of them at a time, depending on the experimental arrangement.  It is not a case of either/or or both/and.  They are complementary aspects of the same physical system.
 Bohr, along with many physicists, would say that it is meaningless to ask which trajectory a quantum particle takes when we are observing an interference pattern.  Since you cannot observe the trajectory at the same time as observing the interference, it simply does not have a trajectory.  Note that in popular science accounts, it is often said that the particle travels along //both// trajectories.  But the conventional view is not //both//, not //either/or//, and not //neither//, but that the particle is in a new kind of physical state where the question itself is meaningless.
 Whether you buy this conventional view is, to some extent, a matter of taste.  There are interpretations of quantum mechanics in which each electron does in fact go through one or the other slit, and interpretations where it does, in some sense, go through both slits.  All we have shown is that you cannot //detect// which slit it goes through without disturbing the interference pattern, and it may be the case that there are things that exist that we cannot directly detect.  Bohr's view of this experiment is optional, although it is the conventional view in the physics community.
+====== 1.vi.3 The Superposition Principle ======
+In quantum mechanics, the //**superposition principle**// states two things:
+  * //Existence of superpositions//: If $\psi_1(\vec{r},t_0)$ and $\psi_2(\vec{r},t_0)$ are solutions to the equations of motion of quantum mechanics at time $t_0$ then so is
+  \[\alpha \psi_1(\vec{r},t_0) + \beta \psi_2(\vec{r},t_0),\]
+  for arbitrary complex coefficients $\alpha$ and $\beta$.
+  * //Preservation of superpositions//: When a system is isolated (not interacting with its environment or being measured) then if, according to the equations of motion, the solution $\psi_1(\vec{r},t_0)$ evolves to $\psi_1(\vec{r},t)$ and the solution $\psi_2(\vec{r},t_0)$ evolves to $\psi_2(\vec{r},t)$ for $t>t_0$, then the solution $\alpha \psi_1(\vec{r},t_0) + \beta \psi_2(\vec{r},t_0)$ evolves to $\alpha \psi_1(\vec{r},t) + \beta \psi_2(\vec{r},t)$.
+The superposition principle is responsible for the wave-like interference effects we have been discussing.  It holds because the equation of motion of quantum mechanics (the Schrödinger equation) is a linear, homogeneous differential equation, just like the wave equation in classical physics.
+{{:question-mark.png?direct&50|}}
+====== In Class Activity ======
+  - In light of the double slit experiment, many physicists are inclined to say either:
+    * Only the wavefunction exists.  Other properties only come into existence when observed.
+    * Nothing exists until observed.
+Consider the experiment depicted below.
+{{ :einsteinsscreen.png?direct&600 |}}
+In this experiment, which principle of physics would be violated by the "only the wavefunction" or "nothing exists" points of view?