What is the classical case of the state of a system?

“State of a system at time t”. precise values of all the dynamical variables at time t.

What is an example of the classical case of the state of a system?

Snooker: white ball has angular momentum and linear momentum, and black ball has centre of mass at (x,y).

What can we say about the classical case of the state of a system?

State evolves according to equations of motion. Evolution of the state is fully deterministic. No fundamental limit on precision of our knowledge of the dynamical variables.

What is the quantum case of the state of a system?

Cannot know accurately certain pairs of dynamical variables e.g. x, p(x) (HUP). Can’t even say that a quantum particle even has a precise but unknown (x, p(x))

What is an example of the quantum case of the state of a system?

Two-slit experiment: Waveform goes to 2 slits which diffracts light onto screen. Creates interference pattern.

For the two-slit example, what is the state of the system before and after measurement?

Before: free particle with precise k(z). After: photon has hit screen at (x, 0)

What is the state of the system in QM defined by?

Its state function (often a wave function).

State the TDSE

iћ *d/dt(Ψ(r,t)) = H*Ψ(r,t), where H = -ћ^2/2m *∇^2 + V(r)

What does the TDSE show us?

How state evolves when no measurement is made.

What is hilbert space?

Like a vector space for functions

How can we represent a state in QM and what is an example of this?

A wave function. Can build functions out of linear combinations of other functions, like Fourier Series.

Suppose we measure a dynamical variable D of state Ψ1(r) and always get D=D1, and Ψ2(r) and always get D=23. What is the equation for Ψ(r)?

Superposition principle: Ψ(r) = c1*Ψ1(r) + c2*Ψ2(r), where c1 and c2 are complex.

What is another example for states in QM?

Polarised light: can plane-polarise a beam of light and pass it through a polarising filter. Transmitted light can only have a polarisation defined by the filter.

What is the equation for light transmitted through polarising filter? What is the equation for the fraction absorbed?

T = cos^2(α), where α is the angle at which the light is polarised. Fraction absorbed is T+A = 1, where A = sin^2(α).

If we consider the intensity of the light being polarised to be one photon, what is the wave function of the incident photon?

Wave func of incident photon is linear superposition of Ψ(x) and Ψ(y) :polaries along x or y.

How do we determine what happened to the photons after measurement (the filter is the measuring device)? What does this mean?

Photons which are transmitted must have Ψ = Ψ(x), and absorbed have Ψ = Ψ(y) : the measurement has changed Ψ - it collapses from superposition to either Ψ(x) or Ψ(y)

What do we write wave functions as?

Superposition of eigenfunctions of a particular operator.

What do the different parts in a measurement represent in QM?

Dynamical variable = operator, result of a measurement = eigenvalue of this operator, wave func after measurement = eigenfunction of this operator

What happens if we twist our filter round by angle β?

x-axis because x’, and measurement results are different: P’(T) = cos^2(α-β), and P’(A) = sin^2(α-β)