Definitions Flashcards
Frequency
The frequency ν of a wave is the number of cycles in one second. Units are Hertz.
Period
The period is T = 1/ν. Units are seconds.
Wavelength
The wavelength λ of a wave is the distance between successive crests. Units are metres.
Wavenumber
The wavenumber is k = 2π/λ. Units are 1/metres (m^-1).
Velocity
The velocity is given by v = λν.
Black Body
A black body emits EM radiation of all wavelengths.
Planck’s Constant
The constant h is known as Planck’s constant. Its value is h = 6.626x10^-34 Js.
Reduced Planck Constant
ℏ = h/2π = 1.0546 x 10^-34 Js. This is known as the reduced Planck Constant.
Photon
A quantum of light.
Work Function
The minimum quantity of energy required to remove an electron from the surface of a given solid, usually a metal.
The Compton Effect
Photons not only have energy, but also momentum. In collisions, photons behave like particles.
Wavelength of EM radiation
The wavelength of the EM radiation with frequency ν is λ = c/ν, where c is the speed of light.
Momentum of a Photon
The momentum of a photon is given by p = h/λ = hν/c.
De Broglie Matter Waves
Proposed matter waves with Wavelength and Frequency, associated with momentum and energy.
De Broglie Wavelength
Any particle with momentum p has an associated wavelength λ = h/p, known as the De Broglie Wavelength.
De Broglie Frequency
If particle has energy E, then the matter wave has frequency ν = E/h.
Wave-Particle Duality
The fact that EM waves can exhibit particle-like behaviour and that matter particles can exhibit wave-like behaviour.
Angular Frequency
ω = 2πν is the angular frequency. ω = 2π/T, where T is period.
Euler’s Formula
e^iθ = cosθ +isinθ
Energy of a De Broglie Wave
E = hν = (h/2π)2πν = ℏω
Momentum of a De Broglie Wave
p = h/λ = ℏk
Energy Operator
The energy operator is iℏ∂/∂t.
Momentum Operator
The momentum operator is -iℏ∂/∂x.
Time-dependent Schrodinger Equation
iℏ∂/∂tψ(x,t) = -(ℏ^2/2m)∂^2/∂x^2ψ(x,t) + V(x)ψ(x,t) where ψ is called the wave function.
Wave Function
A wave function is a mathematical description of the quantum state of an isolated quantum system.
Stationary State
A solution of the TDSE of the form ψ(x ,t ) = exp(−iEt/ℏ)φ(x), is known as a stationary state.
Time-independent Schrodinger Equation
The time-independent Schrodinger equation (TISE) is
−(ℏ^2/2m)d^2φ/dx^2 + V(x)φ = Eφ.
Ground State Energy
The lowest possible energy is called the ground state energy.
Excited State Energies
The higher energies, in increasing order, are called first excited state energy, second excited state energy, …
Quantum Number
The indexing parameter n of energy states is called the quantum number.
Mean Value of the Position
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 .
Mean Value of x^2
The mean value of x^2 is Eψ(x^2) = ∫x^2|ψ(x ,t )|^2 dx .
Variance
The variance is given by [∆ψ(x)]^2 = E_ψ(x^2) − [E_ψ(x)]^2.
Dispersion/Standard Deviation
The dispersion (or standard deviation) is given by
∆ψ(x) = {E_ψ(x^2) − [E_ψ(x)]^2}^1/2.Probability Current
The probability current j(x,t) is defined as
j(x,t) = − iℏ/2m({ ̅ Ψ}∂ψ/∂x − ψ∂{ ̅ Ψ}/∂x).
Normalisable Wave Functions
Normalisable wave functions are called bound states of the system.
Non-normalisable Wave Functions
Non-normalisable wave functions are called scattering states.
Potential Step
V(x) = 0 x < 0
V_0 x > 0
Reflection Coefficient
R = |reflected current|/|incident current| = |j_r|/|j_i|.
Transmission Coefficient
T = |transmitted current|/|incident current| = |j_t|/|j_i|.
Potential Barrier
V(x) = V_0 0 < x < a
0 otherwise
Finite Potential Well
V (x ) = −V_0 0 < x < a
0 otherwise
Schrodinger Equation in 2 Dimensions
iℏ∂/∂t ψ(x,y,t) = −ℏ^2/2m∇^2ψ(x,y,t) + V(x,y)ψ(x,y,t).
2D Laplacian
∇^2 = ∂^2/∂x^2 + ∂^2/∂y^2
Degeneracy
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.
Postulate 1
For a physical quantum system, there exists a state vector (or wave function) ψ which contains all the information about the system.
Postulate 2
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.
Inner Products
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〉.
Complex Inner Product Space
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).
Linear Operator
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.
Adjoint of a Linear Operator
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.
Postulate 3
Each dynamical variable (observable) of a quantum system is described by a self-adjoint linear operator in H.
Hermitian Matrix
M = { ̅M}^T
Eigenvalue and Eigenvector
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.
Normalisable Vector
A vector ψ ∈H is normalisable if ||ψ||^2 =〈ψ|ψ〉= 1,
where〈·|·〉 is an inner product on H.
Expectation Value of an Operator
The expectation value of an operator ˆA for a state ψ is
E_ψ( ˆA) =〈ψ∣ˆAψ〉/‖ψ‖^2.
Dispersion of an Operator
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.
Taylor Series for an Operator
f (ˆA) = ∞∑n=0 f(n)(0)/n! ˆAn = f(0)ˆA^0 + f′(0)ˆA + 1/2!f′′(0) ˆA^2 + …
Commutation of Operators
Two operators ˆA and ˆB commute if ˆAˆB = ˆBˆA.
Commutator
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.
Properties of the Commutator
Antisymmetry, Bilinearity, Product Property, Jacobi Identity
Antisymmetry
[ ˆA,ˆB ] = −[ ˆB ,ˆA].
Bilinearity
[A,γˆB + μˆC] = γ[ˆA,ˆB] + μ[ˆA,ˆC] ;
[γ ˆA + μˆB,ˆC] = γ[ˆA, ˆC] + μ[ˆB,ˆC]
Product Property
[ˆA,ˆBˆC] = ˆB[ˆA,ˆC] + [ˆA,ˆB]ˆC ; [ˆAˆB,ˆC]= ˆA[ˆB,ˆC]+[ˆA,ˆC]ˆB
Jacobi Identity
[ ˆA,[ˆB,ˆC]] + [ˆB ,[ˆC, ˆA]] + [ˆC ,[ˆA,ˆB]] = 0.
Ehrenfest’s Theorem
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 )).
Heisenberg’s Uncertainty Principle
For any state ψ, the dispersion of position ˆx and momentum ˆp satisfy ∆_ψ(ˆx)∆_ψ(ˆp) ≥ ℏ/2.
Ket
It is formatted as | v ⟩ and represents a vector which physically represents a state in a quantum system.
Bra
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.
The Annihilation Operator
The Annihilation Operator is ˆA = sqrt{mω/2ℏ} ˆx + iˆp/sqrt{2mℏω}.
The Creation Operator
The Creation Operator is ˆA* = sqrt{mω/2ℏ}ˆx - iˆp/sqrt{2mℏω}.
The Number Operator
The Number Operator is ˆN = ˆA*ˆA.
