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PHYS1220 – Quantum Mechanics

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PHYS1220 – Quantum Mechanics

Lecture 5

August 28, 2002

Dr J. Quinton

Office: PG 9

ph 49-21-7025

phjsq@alinga.newcastle.edu.au

- slit 1 is open, slit 2 closed

- slit 1 is closed, slit 2 open

- Both slits open

electrons or photons

- “Schrödinger's cat” thought experiment
- A cat is sealed inside a box with air, food, and water to survive.
- This box is designed so that the cat is totally cut off from observation.
- no information (i.e., sight, sound, etc.) can pass into or out of it

- Also inside the box is a Geiger counter; which controls a switch that causes a hammer to automatically break a a glass vial of cyanide gas (which kills the cat) if a radioactive decay is detected.
- The ‘radioactive’ sample is chosen so that after one hour, there is a 50% chance of a decay event.
- Question: What is the state of the cat after one hour has elapsed?

- The intuitive answer is that the cat is either alive or dead, but you don't know which until you look.
- Quantum mechanics - the wave function describing the cat is actually in a superposition of states
- the cat is 50% alive and 50% dead! It is both.
- Not until an observation that "collapses the wave function" to either or is the universe forced to choose between a live cat and a dead cat.
- This thought experiment further indicates that observation seems to be an important part of the scientific process. The ideal of any objective measurement (where the experimenter is independent of the experiment) is not possible in such cases

- Infinitely deep square well potential
- Inside the well, already know the solution
- Outside the well
- Boundary Conditions
- But we don’t want A=0, so

- Energy is quantised (quantum number n)
- Note that the lowest energy a system may have, E1, is not equal to zero. This is purely a quantum phenomenon and is called the zero point energy.
- State wave functions are given by

- Normalisation
- Giancoli eg 39-5

- Question: What is the probability of finding the ground state particle within L/4 of the centre (ie ) of the well?
- Answer: The Probability is given by
- Classically, the probability of finding the particle is the same everywhere in the box, so any region L/2 wide, probability=50%

- Potential in regions I and III
- Schrödinger's equation:
- First consider E<U0. In regions I and III
- the equation has a solution of the form:
- Logically:

- Boundary conditions:
- Well behaved.
- First derivatives equate.

- In regions I and III the wave function has the form of an exponential decay:
- In region II the wavefunction has the form of a sinusoid:
- The boundary conditions allow the constants to be determined:
- eg at x=0
- also 2 conditions for x=L
- 1 from normalisation
- 5 eqns, 5 unknowns

- The mathematics gets rather complicated, so we’ll just focus on the major results in a qualitative sense
- The wave functions and probability densities penetrate the walls with an exponential decay
- features that are purely a quantum mechanics phenomenon
- classically, an object cannot penetrate the barrier

- Case where E>U0
- In all regions the wave function has the form of a sinusoid.
- The potential in region II is different from I and II.
- The boundary conditions allow the wavefunction to be determined in each region, though the system has a continuum in energy.
- Regions I and III have
- Region II has U0=0