"State" of a system and dynamic evolution Flashcards Preview

(2) Quantum Mechanics (Term 1) > "State" of a system and dynamic evolution > Flashcards

Flashcards in "State" of a system and dynamic evolution Deck (19)
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1
Q

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

A

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

2
Q

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

A

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

3
Q

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

A

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.

4
Q

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

A

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))

5
Q

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

A

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

6
Q

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

A

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

7
Q

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

A

Its state function (often a wave function).

8
Q

State the TDSE

A

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

9
Q

What does the TDSE show us?

A

How state evolves when no measurement is made.

10
Q

What is hilbert space?

A

Like a vector space for functions

11
Q

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

A

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

12
Q

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)?

A

Superposition principle: Ψ(r) = c1Ψ1(r) + c2Ψ2(r), where c1 and c2 are complex.

13
Q

What is another example for states in QM?

A

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.

14
Q

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

A

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).

15
Q

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

A

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

16
Q

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

A

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)

17
Q

What do we write wave functions as?

A

Superposition of eigenfunctions of a particular operator.

18
Q

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

A

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

19
Q

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

A

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