quantum mechanics - 2074 exam 2

what information about a particle (or a system of particles) is required in order for its future (and past) to be determined?

position and velocity (momentum) at a given instant

which quantity (if any) contains all the information we can possibly know about a particle?

its position as a function of time (r(t)); if we know it we can calculate any other quantity such as velocity, momentum, energy, etc

what additional element is required in order for us to to be able to predict the future of the particle?

infinite precision in measuring position and velocity as well as infinite computing capacity

name properties shared by waves that are not found in particles

interference (constructive/destructive), diffraction

name properties shared by particles not found in waves

mass, localization, indivisibility

give some examples of waves

waves on a stretched string (1D), waves on the surface of a body of water or a drum membrane (2D), sound, light, gravitational waves (3D or higher)

give some examples of particles

atoms (only in CM), a speck of dust, a baseball, a planet

what is a good test to see whether something is a stream of particles or a wave?

double slit experiment

name some phenomena/experiments that do not make sense classically (assuming particles are localized points of mass and light is a wave)

atomic stability, blackbody radiation spectrum, photoelectric effect, radioactivity, double slit experiment using electrons

what is a possible way out of the difficulty of things not making sense classically?

postulate that CM does not apply to microscopic particles

what is the definition of of the probability of an event?

the ratio of the number of times that the given event is observed by the total number of trials

what properties must probabilities satisfy?

non-negative, between 0 and 1, all of them together must add up to 1

what is an ensemble?

a large (infinite) number of identical systems

state two important statistical quantities

average and standard deviation of a set of measurements

what is the intuitive meaning of standard deviation

the spread of the measurements

what quantity contains all the information that we can possibly know about a particle/system?

its wavefunction

what is one way to obtain the wavefunction for a particle/system (in principle)?

by solving the Schrodinger equation

on how many things does a wave function depend?

position and time

what does the square of a wavefunction tell us about a particle?

the probability of locating the particle near a given point at a given time (sometimes referred to as the Copenhagen interpretation)

if we know the wavefunction of the particle can we calculate the probabilities of other measureable quantities?

in principle yes (all information is contained in the wavefunction)

given the graph of a wavefunction how can we find the corresponding probability of locating the particle near a given point?

square the wavefunction (no negative values for a probability)

does the uncertainty principle state that “everything is uncertain”?

no, it states that the more we know about the position of a particle (standard deviation of position measurements is small), the less we know about its velocity/momentum (SD of these measurements is large); taking to an extreme, if we know EXACTLY where a particle is, then any value for a velocity becomes equally probable as any other value

where was the particle before we measured its position?

in QM, if the particle was exactly there before the measurement was made, then the theory is deficient because it could only provide a probability of finding the particle there and not a certainty

what is the “realist” view for QM?

if the particle was exactly there before the measurement was made, then it could only provide a probability (not a certainty) of finding a particle there because other factors (hidden variables) are missing in order to make the theory complete and yield unique predictions 

what is the “orthodox” view for QM?

the particle was not anywhere before the measurement was made because the act of measurement is what produced the particle as an indivisible unit

where is the particle immediately after its position is measured?

the particle is exactly at the location where it was found because the measurement of position makes its wavefunction collapse to a spike causing us to lose all information about the particle’s velocity/momentum

what about all other possible measurement outcomes that do not happen to be observed when a measurement takes place?

one view is that they all exist in different universes that are unobservable to us (many worlds view)

why do we observe quantum behavior in our everyday experience of macroscopic objects (realm of classical mechanics)?

macroscopic objects consist of a very large number of microscopic particles, in which particles constantly interact with other particles; this interaction can be called a measurement, which causes a wavefunction to collapse into what we may call a more intuitive particle with a given location

what is the EPR experiment?

a suggested experiment to support the claim that QM although useful is not a complete theory