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MAS 324 Quantum Theory > Definitions > Flashcards

Definitions Flashcards

1
Q

Frequency

A

The frequency ν of a wave is the number of cycles in one second. Units are Hertz.

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2
Q

Period

A

The period is T = 1/ν. Units are seconds.

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3
Q

Wavelength

A

The wavelength λ of a wave is the distance between successive crests. Units are metres.

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4
Q

Wavenumber

A

The wavenumber is k = 2π/λ. Units are 1/metres (m^-1).

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5
Q

Velocity

A

The velocity is given by v = λν.

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6
Q

Black Body

A

A black body emits EM radiation of all wavelengths.

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7
Q

Planck’s Constant

A

The constant h is known as Planck’s constant. Its value is h = 6.626x10^-34 Js.

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8
Q

Reduced Planck Constant

A

ℏ = h/2π = 1.0546 x 10^-34 Js. This is known as the reduced Planck Constant.

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9
Q

Photon

A

A quantum of light.

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10
Q

Work Function

A

The minimum quantity of energy required to remove an electron from the surface of a given solid, usually a metal.

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11
Q

The Compton Effect

A

Photons not only have energy, but also momentum. In collisions, photons behave like particles.

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12
Q

Wavelength of EM radiation

A

The wavelength of the EM radiation with frequency ν is λ = c/ν, where c is the speed of light.

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13
Q

Momentum of a Photon

A

The momentum of a photon is given by p = h/λ = hν/c.

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14
Q

De Broglie Matter Waves

A

Proposed matter waves with Wavelength and Frequency, associated with momentum and energy.

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15
Q

De Broglie Wavelength

A

Any particle with momentum p has an associated wavelength λ = h/p, known as the De Broglie Wavelength.

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16
Q

De Broglie Frequency

A

If particle has energy E, then the matter wave has frequency ν = E/h.

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17
Q

Wave-Particle Duality

A

The fact that EM waves can exhibit particle-like behaviour and that matter particles can exhibit wave-like behaviour.

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18
Q

Angular Frequency

A

ω = 2πν is the angular frequency. ω = 2π/T, where T is period.

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19
Q

Euler’s Formula

A

e^iθ = cosθ +isinθ

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20
Q

Energy of a De Broglie Wave

A

E = hν = (h/2π)2πν = ℏω

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21
Q

Momentum of a De Broglie Wave

A

p = h/λ = ℏk

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22
Q

Energy Operator

A

The energy operator is iℏ∂/∂t.

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23
Q

Momentum Operator

A

The momentum operator is -iℏ∂/∂x.

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24
Q

Time-dependent Schrodinger Equation

A

iℏ∂/∂tψ(x,t) = -(ℏ^2/2m)∂^2/∂x^2ψ(x,t) + V(x)ψ(x,t) where ψ is called the wave function.

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25
Q

Wave Function

A

A wave function is a mathematical description of the quantum state of an isolated quantum system.

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26
Q

Stationary State

A

A solution of the TDSE of the form ψ(x ,t ) = exp(−iEt/ℏ)φ(x), is known as a stationary state.

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27
Q

Time-independent Schrodinger Equation

A

The time-independent Schrodinger equation (TISE) is

−(ℏ^2/2m)d^2φ/dx^2 + V(x)φ = Eφ.

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28
Q

Ground State Energy

A

The lowest possible energy is called the ground state energy.

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29
Q

Excited State Energies

A

The higher energies, in increasing order, are called first excited state energy, second excited state energy, …

30
Q

Quantum Number

A

The indexing parameter n of energy states is called the quantum number.

31
Q

Mean Value of the Position

A

The mean value (or expectation value) of the position x of the particle is given by E_ψ(x) = ∫x^2|ψ(x,t)|^2 dx .

32
Q

Mean Value of x^2

A

The mean value of x^2 is Eψ(x^2) = ∫x^2|ψ(x ,t )|^2 dx .

33
Q

Variance

A

The variance is given by [∆ψ(x)]^2 = E_ψ(x^2) − [E_ψ(x)]^2.

34
Q

Dispersion/Standard Deviation

A 
The dispersion (or standard deviation) is given by
∆ψ(x) = {E_ψ(x^2) − [E_ψ(x)]^2}^1/2.
35
Q

Probability Current

A

The probability current j(x,t) is defined as

j(x,t) = − iℏ/2m({ ̅ Ψ}∂ψ/∂x − ψ∂{ ̅ Ψ}/∂x).

36
Q

Normalisable Wave Functions

A

Normalisable wave functions are called bound states of the system.

37
Q

Non-normalisable Wave Functions

A

Non-normalisable wave functions are called scattering states.

38
Q

Potential Step

A

V(x) = 0 x < 0

V_0 x > 0

39
Q

Reflection Coefficient

A

R = |reflected current|/|incident current| = |j_r|/|j_i|.

40
Q

Transmission Coefficient

A

T = |transmitted current|/|incident current| = |j_t|/|j_i|.

41
Q

Potential Barrier

A

V(x) = V_0 0 < x < a

0 otherwise

42
Q

Finite Potential Well

A

V (x ) = −V_0 0 < x < a

0 otherwise

43
Q

Schrodinger Equation in 2 Dimensions

A

iℏ∂/∂t ψ(x,y,t) = −ℏ^2/2m∇^2ψ(x,y,t) + V(x,y)ψ(x,y,t).

44
Q

2D Laplacian

A

∇^2 = ∂^2/∂x^2 + ∂^2/∂y^2

45
Q

Degeneracy

A

If the space of solutions of Schrodinger’s time-independent equation with energy E has dimension k > 1, then the energy level is k-fold degenerate or has degeneracy k. If the space is one-dimensional, then the energy is non-degenerate.

46
Q

Postulate 1

A

For a physical quantum system, there exists a state vector (or wave function) ψ which contains all the information about the system.

47
Q

Postulate 2

A

If ψ_1 and ψ_2 are state vectors of a particular system, then a linear combination c_1ψ_1 + c_2ψ_2 is also a possible state vector, where c_1 and c_2 are arbitrary complex numbers.

48
Q

Inner Products

A

For two wave functions ψ and φ, their inner product is defined to be〈φ|ψ〉=∫{ ̅φ}ψ dx, where { ̅φ} denotes the complex conjugate of φ. The inner product associates each pair of complex wave functions ψ and φ with a complex scalar quantity 〈φ|ψ〉.

49
Q

Complex Inner Product Space

A

States of a quantum system are described by non-zero vectors in acomplex inner product space (or Hilbert space) H, endowed with an inner product 〈φ|ψ〉.
Two vectors in H describe the same state if and only if they are multiples (ψ ↔ e^{iθ}ψ).

50
Q

Linear Operator

A

A linear operator ˆA satisfies ˆA(c_1ψ_1 + c_2ψ_2) = c_1 ˆAψ_1 + c_2 ˆAψ_2, for all ψ_1, ψ_2 ∈H and c_1, c_2 ∈ C.

51
Q

Adjoint of a Linear Operator

A

The adjoint of a linear operator ˆA is the unique operator ˆA∗ such that 〈ˆA∗φ∣ψ〉=〈φ∣ˆAψ〉 for all ψ,φ ∈H. An operator ˆA is self-adjoint if ˆA∗ = ˆA.

52
Q

Postulate 3

A

Each dynamical variable (observable) of a quantum system is described by a self-adjoint linear operator in H.

53
Q

Hermitian Matrix

A

M = { ̅M}^T

54
Q

Eigenvalue and Eigenvector

A

For a linear operator ˆA, if there is a numberλ∈C and a vector ψ∈H such that ψ ≠ 0 and ˆAψ = λψ, then λ is an eigenvalue of ˆA and ψ is the corresponding eigenvector.

55
Q

Normalisable Vector

A

A vector ψ ∈H is normalisable if ||ψ||^2 =〈ψ|ψ〉= 1,

where〈·|·〉 is an inner product on H.

56
Q

Expectation Value of an Operator

A

The expectation value of an operator ˆA for a state ψ is

E_ψ( ˆA) =〈ψ∣ˆAψ〉/‖ψ‖^2.

57
Q

Dispersion of an Operator

A

The dispersion (or standard deviation) of an operator ˆA in a state ψ is∆ψ( ˆA) ={E_ψ( ˆA^2) − [Eψ( ˆA)]^2}^1/2.

58
Q

Taylor Series for an Operator

A

f (ˆA) = ∞∑n=0 f(n)(0)/n! ˆAn = f(0)ˆA^0 + f′(0)ˆA + 1/2!f′′(0) ˆA^2 + …

59
Q

Commutation of Operators

A

Two operators ˆA and ˆB commute if ˆAˆB = ˆBˆA.

60
Q

Commutator

A

The commutator of two operators ˆA and ˆB is defined as [ˆA,ˆB] = ˆAˆB − ˆBˆA. Two operators commute if and only if their commutator vanishes.

61
Q

Properties of the Commutator

A

Antisymmetry, Bilinearity, Product Property, Jacobi Identity

62
Q

Antisymmetry

A

[ ˆA,ˆB ] = −[ ˆB ,ˆA].

63
Q

Bilinearity

A

[A,γˆB + μˆC] = γ[ˆA,ˆB] + μ[ˆA,ˆC] ;

[γ ˆA + μˆB,ˆC] = γ[ˆA, ˆC] + μ[ˆB,ˆC]

64
Q

Product Property

A 
[ˆA,ˆBˆC] = ˆB[ˆA,ˆC] + [ˆA,ˆB]ˆC ; 
[ˆAˆB,ˆC]= ˆA[ˆB,ˆC]+[ˆA,ˆC]ˆB
65
Q

Jacobi Identity

A

[ ˆA,[ˆB,ˆC]] + [ˆB ,[ˆC, ˆA]] + [ˆC ,[ˆA,ˆB]] = 0.

66
Q

Ehrenfest’s Theorem

A

For a Hamiltonian ˆH = ˆp^2/2m+ ˆV and state ψ, the expectation values of position and momentum satisfy
md/dt E_ψ(ˆx ) = E_ψ(ˆp), d/dt E_ψ( ˆp) = −E_ψ(V ′(ˆx )).

67
Q

Heisenberg’s Uncertainty Principle

A

For any state ψ, the dispersion of position ˆx and momentum ˆp satisfy ∆_ψ(ˆx)∆_ψ(ˆp) ≥ ℏ/2.

68
Q

Ket

A

It is formatted as | v ⟩ and represents a vector which physically represents a state in a quantum system.

69
Q

Bra

A

It is formatted as ⟨ f | , and mathematically denotes a linear form f : V → C , i.e. a linear map that maps each vector in V to a number in the complex plane C Letting the linear functional ⟨ f | act on a vector | v ⟩ is written as ⟨ f | v ⟩ ∈ C.

70
Q

The Annihilation Operator

A

The Annihilation Operator is ˆA = sqrt{mω/2ℏ} ˆx + iˆp/sqrt{2mℏω}.

71
Q

The Creation Operator

A

The Creation Operator is ˆA* = sqrt{mω/2ℏ}ˆx - iˆp/sqrt{2mℏω}.

72
Q

The Number Operator

A

The Number Operator is ˆN = ˆA*ˆA.

MAS 324 Quantum Theory (1 decks)

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