Physics 2170 – Spring 2009 1http://www.colorado.edu/physics/phys2170/

Review 1

• Last homework is due tomorrow at 12:50pm• Final exam is on Saturday from 1:30pm – 4:00 pm in G125

(this room)• I should be in my office from 2-5pm Thursday and 9-12

and 2-5 on Friday if you have questions– You can also send me email

• The formula sheet for the final is basically the first two formula sheets put together plus a few more formulae. It will be put up by 4pm today.

• Today is the last day clickers count although there will be clicker questions Friday.

• Cover relativity today and quantum mechanics Friday.

Announcements:

Physics 2170 – Spring 2009 2http://www.colorado.edu/physics/phys2170/

Final exam• It is about 1/3 relativity and the rest is quantum

mechanics with an emphasis on the last few weeks– There are 3 multipart problems and a total of 25

questions.– Problems are similar to midterms – maybe a little less

involved.• I recommend studying the following:

– Old finals on CULearn (don’t worry about questions on semiconductors)

• I think these finals are a little easy; don’t expect your final to be that easy.

• Try to take the old finals as if you were taking a real test.– The 14 homework assignments– The lectures (including the clicker questions)– The midterms

Physics 2170 – Spring 2009 3http://www.colorado.edu/physics/phys2170/

Schrödinger’s CatA radioactive sample has a 50% chance of emitting an alpha particle. If it decays, a Geiger detector triggers the release of poison killing a cat in the box. Before opening the box, the cat is in a superposition of wave functions:

alive21

dead21

When does the wave function collapse to either dead or alive?

No clear agreement. Interesting physics/philosophical question.

Physics 2170 – Spring 2009 4http://www.colorado.edu/physics/phys2170/

Quantum entanglement/teleportationSuppose a particle with angular momentum of 0 decays into two electrons. By conservation of angular momentum, the sum of the two electrons angular momentum must also be 0.

If we measure the z-spin of one electron to be +½ then we know that the other electron must have a z-spin of –½. But before we measure the first electron, it is in a mixture of +½ and –½ spin states. The act of measuring causes the electron to have a definite spin.

We can separate the two electrons, measure the 1st electron and then measure the 2nd electron before any possible (light speed) signal can reach the 2nd electron. And yet the 2nd electron always has the correct spin.

Einstein called this “spooky action at a distance”

Physics 2170 – Spring 2009 5http://www.colorado.edu/physics/phys2170/

Quantum entanglement/teleportationIn 1935 Einstein, Podolsky, and Rosen wrote a paper attempting to show that quantum mechanics results in a paradox (now called the EPR paradox).

In 1964, J.S Bell proved that local hidden variable theories would give a different result in some cases than quantum mechanics.

They proposed a way out: Electrons actually always know their spin in every direction but experiments can only get the limited knowledge allowed by quantum mechanics. A better theory would allow one to get access to this information.

This is called a hidden variable theory.

Experiments in the 70s & 80s confirmed that quantum mechanics was correct and local hidden variable theories don’t work.

Physics 2170 – Spring 2009 6http://www.colorado.edu/physics/phys2170/

Special Relativity – Chapter 1Postulates of relativity:

1. All physical laws are the same in all inertial reference frames

2. The speed of light is the same in all inertial reference frames

Proper time t0 between two events is the time measured in the frame in which both events occur at the same location.

Time dilation (moving clocks run slower): t =t0

Length contraction (moving objects are shorter in the direction they are traveling): L = L0/

Relativity of simultaneity: events which are simultaneous in one frame may not be in another; even the order may be switched.

Proper length L0 of an object is the length measured in the rest frame of the object.

Physics 2170 – Spring 2009 7http://www.colorado.edu/physics/phys2170/

Set frequency to DAClicker question 1Jimmy sees two lightning flashes and determines that one happens 10 ns before the other. Which statement is true:A. Another inertial observer must find the order of events, the

location of events, and the time difference to be the same.B. Another inertial observer must find the order of events to be the

same but the location and time difference may be differentC. Another inertial observer must find the order of events and the

locations to be the same but the time difference may be different.D. Any other inertial observer might find the order of events, the

locations, and the time difference to be different.E. Say what?

In general, length, time, and order can be different for different inertial observers.

However, due to causality and finite speed of light, if the events are far enough apart it is impossible for the order to be reversed.

Physics 2170 – Spring 2009 8http://www.colorado.edu/physics/phys2170/

Special Relativity – Chapter 1 (continued)Lorentz transformations are used to transform coordinates (position and time) of an event from a rest frame S to a moving frame S′.

)/(' ' ' )( 2cvxttzzyyvtxx

Velocity addition formula is also different than the Galilean velocity addition formula: 2/1 cuv

vuu

Spacetime diagrams can be used to understand where & when events occur and how they are related in the space-time continuum.

The inverse Lorentz transformations go the other way (S′ to S) and can be obtained by swapping primes and signs.

x

ctFuture(forward

light cone)

Elsewhere

Past (backward light cone)

11

sourceobs ff

Relativistic Doppler shift for observer and source approaching at .

Physics 2170 – Spring 2009 9http://www.colorado.edu/physics/phys2170/

Solving for v:

To get

Set frequency to DAClicker question 2Two events take place 90 m apart with an intervening time interval of 0.60 s in reference frame S. What is the speed of the reference frame S′ which measures the proper time between the two events?

A. 0B. 0.25cC.0.50cD.0.90cE. c

Proper time t0 between two events is the time measured in the frame in which both events occur at the same location.

So in the S′ frame, both events must occur at point 0. That is, x′ = 0 when x = 90 m and t = 0.60 s.

)(

)(

2 xcvtt

zzyy

vtxx

0)( vtxx requires x – vt = 0, i.e. x = vt.

ctxv 5.0sm/ 150

s .6m 90

Physics 2170 – Spring 2009 10http://www.colorado.edu/physics/phys2170/

Set frequency to DAClicker question 3A person on Earth sees two spaceships heading toward each other. The earthling sees rocket A moving at 0.8c. In the Earth frame, how fast are the two spaceships moving toward each other?

A. 0

B. 0.6c

C.0.8c

D.0.98c

E. 1.6c

2/1 cvuvuu

2/1 cuvvuu

The person on Earth sees that they are closing at 1.6c. This does not mean that they see anything moving faster than the speed of light. It is just that they are approaching each other at that speed.

Physics 2170 – Spring 2009 11http://www.colorado.edu/physics/phys2170/

Set frequency to DAClicker question 4A person on Earth sees two spaceships heading toward each other. The earthling measures ship A moving at 0.6c and ship B at 0.8c. In the reference frame of ship A, how fast is ship B moving?

A. 0

B. 0.6c

C.0.8c

D.0.95c

E. 1.6c

2/1 cuvvuu

2/1 cvuvuu

u is the speed of something in the S frame.

u′ is the speed of the same thing in the S′ frame.

v is the relative speed between the S and S′ frames.

We know u: the speed of rocket B in the Earth frame (S frame)

We know v: the relative speed between the S frame (Earth) and the S′ frame (rocket A)

So we need u′: the speed of rocket B in the S′ frame

cccvuvuu 95.0

48.014.1

/1 2

Physics 2170 – Spring 2009 12http://www.colorado.edu/physics/phys2170/

Energy-momentum relationship:

Define kinetic energy:

Special Relativity – Chapter 2Conservation of momentum and energy continues to hold for isolated systems if we redefine momentum and energy as:

mup u2mcE u for mass m with velocity u and 22 /1/1 cuu

222rest )1( mcmcmcEEKE uu

This gives a rest energy of 2rest mcE

2222 )()( mcpcE For a massless particle like a photon, this reduces to pcE

One other equation is Epc /This is it for equations in Chapter 2. Usually use conservation of energy and conservation of momentum to solve problems.

Physics 2170 – Spring 2009 13http://www.colorado.edu/physics/phys2170/

Set frequency to DAClicker question 5A particle with mass 1116 MeV/c2 decays at rest to a proton with mass 938 MeV/c2 and a pion with mass 140 MeV/c2. What can we say about the proton and pion momentum and energy?

A. The proton and pion have the same magnitude momentum and energy.

B. The proton and pion have the same energy but the proton has more momentum.

C. The proton and pion have the same momentum but the proton has more energy.

D. The proton and pion have equal and opposite momenta and the proton has more energy.

E. Need more informationConservation of momentum requires the proton and pion momentum add to 0 so must be equal and opposite. Since E2 = (pc)2 + (mc2)2, the proton has more energy due to larger mass.

2222 )()( mcpcE

2rest mcE

mup u

2mcE u

2rest )1( mcEEKE u

Epc /