Quantum Physics Comes of Age I incident II transmitted reflected

Dispute about the interpretation of QM Einstein Bohr ”God does NOT play dice” ”God plays dice” Dispute about the interpretation of QM

Copenhagen school Bohr, Heisenberg, Pauli

Erwin Schrödinger “The task is not so much to see what no-one has yet seen, but to think what nobody has yet thought, about that which everybody sees.”

Quantum Mechanics: Schrodinger's Cat: History First arose in 1935 Described in letters exchanged between Schrodinger and Einstein  Both were rallying against the Copenhagen Interpretation where "wave function collapse" occurs only at observation Einstein considered a power keg that is paradoxically both exploded and un-exploded Schrodinger agreed, and replied by describing the famous cat paradox Schrodinger did not expect that the experiment would ever be done -- it was considered a "reduction to absurdity" argument against the Copenhagen Interpretation of quantum mechanics

The Thought Experiment A steel chamber

The Thought Experiment (contd.) A cat is placed in it

The Thought Experiment (contd.) And with it, a small amount of a radioactive substance

The Thought Experiment (contd.) And a veil of poison The box is shielded against quantum decoherence

The Thought Experiment (contd.) The cat can be alive 

The Thought Experiment (contd.) The cat can be dead 

The Thought Experiment (contd.) Or both?

The Thought Experiment (contd.) Or both? Yes 

How ?? Any atom in the radioactive substance decays The veil breaks open and releases poison The cat dies

How ?? No atom in the radioactive substance decays The poison is not released The cat is alive

How ?? Atom in the radioactive substance is in a state of both ‘decayed’ and ‘undecayed’ The cat is both dead and alive

The Principle of Superposition The principle of superposition says that if an object can be in any configuration, and if the object could also be in another configuration, then the object can also be in a state which is a superposition of the two. The thought experiment uses this and transforms an indeterminacy in the atomic domain to macroscopic indeterminacy.

The Problem Thus an atom of the radioactive substance can be in both decayed and non-decayed states at the same time. This leads to the conclusion that the cat can be both dead and alive at the same time. But we can never observe the cat to be both dead and alive at the same time!!!

an interesting explanation - The Copenhagen Interpretation A system stops being in a superposition of states and becomes either one or the other when an observation takes place.

Quantum Mechanics: Schrodinger's Cat: Description Cat in a closed box  A quantum decision is made in the box that may kill the cat Time passes

Quantum Mechanics: Schrodinger's Cat: Description Just before the box is opened: is the cat dead? 1. Either yes or no, but you won't know until the box is opened. 2. Both yes and no until the box is opened, then either yes or no.

Schrodinger's Cat: Copenhagen Interpretation 2.  Both yes and no until the box is opened, then either yes or no. The cat is BOTH alive and dead until the box is opened! Cat's "wave function state" collapses only when the box is opened "Opening the box" can really mean actually opening the box looking into the box doing any determinative experiment to the closed box

Schrodinger's Cat Contrasting Interpretations 1.  Either yes or no, but you won't know until the box is opened. Many Worlds Interpretation separate universes house dead and alive cats these universes are decoherent -- do not interact Ensemble Interpretation individual cats are either alive or dead, not both, but you can't know which until the box is opened statistics are only built up when many single-cat systems are observed

Heisenberg’s Matrix Mechanics The Schrödinger Equation Heisenberg’s Matrix Mechanics 1924: de Broglie suggests particles are waves Mid-1925: Werner Heisenberg introduces Matrix Mechanics Semi-philosophical, it only considers observable quantities It used matrices, which were not that familiar at the time It refused to discuss what happens between measurements In 1927 he derives uncertainty principles Late 1925: Erwin Schrödinger proposes wave mechanics Used waves, more familiar to scientists at the time Initially, Heisenberg’s and Schrödinger’s formulations were competing Eventually, Schrödinger showed they were equivalent; different descriptions which produced the same predictions Both formulations are used today, but Schrödinger is easier to understand

The Free Schrödinger Equation 1925: Erwin Schrödinger proposes wave mechanics Peter Debye suggested to him he needed to find a wave equation for quantum mechanics He hit on the idea of using complex waves The rest is history Starting point: Energy/Momentum relationship Multiply by the wave function on the right Use de Broglie relations to rewrite Use relationships for complex waves to rewrite with derivatives

Sample Problem with Free Schrödinger Show that the following expression satisfies the free Schrödinger equation, and find the constant A:

Sample Problem with Free Schrödinger (2) Show that the following expression satisfies the free Schrödinger equation, and find the constant A: Multiply by 2mt5/2/

The Schrödinger Equation The General Prescription for Classical  Quantum: Write a formula for the energy in terms of momentum and position Transform Energy and momentum using the following prescription: Rewrite it as a wave equation What if we have forces? Need to add potential energy V(x,t) on top of kinetic energy term

Comments on Schrödinger Equation 1. This equation is inherently complex You MUST use complex wave functions 2. This equation is first order in time It has only first derivatives with respect to time If you know the value at t = 0, you can work it out at subsequent times Proved using Taylor expansion: Initial conditions: Classical physics x(t = 0) and v(t = 0) Initial conditions: Quantum physics (x,t = 0)

The Superposition Principle 3. This equation is linear The wave function appears to the first power everywhere You can take linear combinations of solutions: Let 1 and 2 be two solutions of Schrödinger. Then so is where c1 and c2 are arbitrary complex numbers Q.E.D

Time Independent problems Often [usually] the potential does not depend on time: V = V(x). To solve this equation, we try separation of variable: Plug this guess in: Divide by the original wave function Note that left side is independent of x, and right side is independent of t. Both sides must be independent of both x and t Both sides must be equal to a constant, called E (the energy)

Solving the time equation We have turned one equation into two But the two equations are now ordinary differential equations Furthermore, the first equation is easy to solve: These types of solutions are called stationary states Why? Don’t they have time in them? The probability density is independent of time

The Time Independent Schrödinger Eqn Multiply by (x) again This equation is much easier to solve than the original equation ODE’s are easier than PDE’s It can pretty easily be solved numerically, if necessary Note that it is a real equation – you don’t need complex numbers Imagine finding all possible solutions n(x) with energy En Then we can find solutions of the original Schrödinger Equation The most general solution is superposition of this solution