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Quantum devices > Semester 2 > Flashcards

Semester 2 Flashcards

1
Q

If ψn(x) is a solution to the time independent schrodinger equation, give an expression for

A

Ψn(x,t) = ψn(x)f(t)

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2
Q

What is the physical meaning of the Hamiltonian?

A

It’s an operator corresponding to the total energy of the quantum system.

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3
Q

If you measure an observable Q on a particle in the state |ψ>, what are you certain to obtain?

A

One of the eigenvalues of the Hermitian operator Q.

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4
Q

How do you prove two vectors form an orthonormal basis?

A

Show that their inner product is equal to zero.

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5
Q

How is a quantum system represented in quantum mechanics?

A

By a wavefunction ψ(x,t)

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6
Q

What is the probability density of finding a quantum particle at position x?

A

P(x) = | ψ(x,t) |^2 dx

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7
Q

What does the normalisation condition allow you to determine?

A

The amplitude of the wavefunction because the integral of the absolute square of the wavefunction over all space must be equal to 1.

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8
Q

What is the average value known as in quantum mechanics?

A

The expectation value

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9
Q

What is wave mechanics?

A

The elementary formulation of quantum theory that is centred on the particles wave function

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10
Q

What is the evolution of the wave function ψ governed by in one dimensional space?

A

The Schrodinger equation

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11
Q

What is a useful way to simplify the schrodinger equation?

A

Looking for separable solutions, with everything dependent on t on one side and everything dependent on x on the other side.

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12
Q

How do you use separable solutions to simplify the schrodinger equation?

A

Substitute Ψ(x,t) for ψ(x)f(t)

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13
Q

What do solutions of the time independent schrodinger equation describe?

A

States of a particle with a definite energy known as stationary states.

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14
Q

What is “I”?

A

I = √-1

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15
Q

What is the wavelength defined as?

A

k = 2π/λ

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16
Q

What is the period defined as?

A

ω = 2π/T

where ω is the angular frequency

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17
Q

What is a time independent wave called?

A

A standing wave

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18
Q

What is a plane wave?

A

A wave that combines space and time oscillations to make a travelling wave f(x,t).

19
Q

What is the intensity of a travelling wave f(x,t)?

A

|f|^2 = f*f = 1

20
Q

What are the four dimensions of a plane wave?

A
  • Real amplitude
  • Imaginary amplitude
  • x
  • t
21
Q

Why is a plane wave difficult to plot and how can this be solved?

A

Because it has four dimensions, to solve this you can make plots of planes for a given phase.

22
Q

How do you switch between domains of time and frequency for a wavefunction?

A

Through fourier transforms

23
Q

What are the three alternate ways of picturing the delta function?

A
  • The limit of a sequence of functions.
  • The derivative of the step function.
  • The sifting functional.
24
Q

What is meant by “free particle”?

A

A moving particle with no forces acting on it, in any region of space, so V (x) is constant for all of x.

25
Q

Why is the energy of the free particle not quantised?

A

Because there are no restrictions on the value of k, which means there are no restrictions on the value of E.

26
Q

What do the first and second terms of the free particle represent?

A

The first term represents a plane wave travelling to the right.

The second term represents a plane wave travelling to the left.

27
Q

Why can a free particle not exist in a stationary state?

A

Because the integral of the wavefunction diverges meaning it cant be normalised. This means there is no such thing as a free particle with a definite energy.

28
Q

What is the De Broglie rule?

A

λ = 2π/k = h/p

29
Q

What is the Einstein relation for energy?

A

E = hc/λ = hω (wave aspect)

30
Q

What is the Einstein relation for a free non-relativistic particle?

A

E = p^2/2m (particle aspect)

31
Q

What is a wave packet?

A

A wave packet can be thought of as a wave which is modulated by an envelope function.

32
Q

What material can be used to create a quantum well?

A

A semiconductor heterostructure such as GaAs or AlGaAs

33
Q

How can you observe transitions between energy levels in a quantum well?

A

Optical spectroscopy

34
Q

What is the current for a stationary state where Ψ(x) is purely real?

A

0

35
Q

What is the superposition of states needed for?

A

To carry a current

36
Q

What do fixed boundary conditions in quantum mechanics result in?

A

Discrete energy levels and wave functions which are standing waves and carry no current

37
Q

What are Born-Von Karman boundary conditions?

A

Periodic boundary conditions that involve repeating the system periodically with the same wave function in each system.

38
Q

What are the allowed values of k for the normalised Born Von-Karman wavefunction?

A

kn = 2πn/L

where n must be an integer

39
Q

In the normalised Born Von-Karman wavefunction, what does each value of k correspond to?

A

A plane wave and a discrete electron wavefunction with a characteristic wavefunction.

40
Q

How can you count the number of electron states in a material?

A

Through knowing the separation of states in k-space

41
Q

What is the density of states function used for?

A

To describe quantum systems

42
Q

Why is the density of states function g(E) used to describe quantum systems?

A

Because a complete description of a system requires the energies and wavefunctions of all its states, and this is impossible for all but the simplest physical systems.

43
Q

What is the definition of the density of states function?

A

g(E)δ(E) is the number of states in the system whose energies lie in the range of E to E + δE

Quantum devices (3 decks)

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