Quantum Theory Flashcards
Give 5 problems which are unresolved without quantum theory
- Photoelectric effect
- Discrete emission/absorption spectra
- Double slit experiment
- De-Broglie’s matter-wave properties
- Compton scattering
Give the de Broglie relations
E = ℏω, p = ℏk, where E is the energy of a free particle, p is the momentum associated with a wave of angular frequency ω and k is the wave vector.
State Schrodinger’s Equation
iℏ(∂Ψ/∂t) = −ℏ^2(∇^2)Ψ/2m + VΨ
State the Stationary State Schrodinger Equation
−ℏ^2(∇^2)ψ/2m + Vψ = Eψ, where Ψ(x,t) = ψ(x)e^(−iEt/ℏ)
For a particle with potential 0 inside the interval (0,a) and infinite potential outside, give the permitted energies of the particle.
E = En = (nπℏ)^2/2ma^2.
Define the ground state
When the possible energies of a system are discrete and bounded below, the wavefunction with the lowest possible energy is called the ground state.
For a particle with potential 0 inside the interval (0,a)x(0,b)x(0,c) and infinite potential outside, give the permitted energies of the particle.
E = E_(n1,n2,n3) = (πℏ)^2/2m[(n1/a)^2+(n2/b)^2+(n3/c)^2]
Define degeneracy
The degeneracy of an energy level E of a system is the dimension of the solution space to the stationary state Schrödinger equation with energy E (though by convention, if this is one, it is said to be non-degenerate).
Give the probability postulate
The function ρ(x, t) ≡ |Ψ(x, t)|^2 is a probability density function for the position of the particle, where Ψ(x, t) is the particle’s wave function.
Define a normalisable wave function
A wave function Ψ is normalisable if 0 < triple integral over R3 of |Ψ(x, t)|^2 < ∞ for all t.
Define a normalised wave function
A wave function Ψ is normalised if triple integral over R3 of |Ψ(x, t)|^2 =1 for all t.
Define the correspondence principle
The tendency for quantum results to approach classical results for large quantum numbers.
Define the expectation of a position function
The expectation value of a function of position f(x) is given by the triple integral over R3 of f(x)|Ψ(x, t)|^2
Define the probability current of a wave function.
j(x,t) = iℏ/2m[Ψ(x,t){(∇Ψ)(x,t)}* − {Ψ(x,t)}*(∇Ψ)(x,t)].
Give the conditions of a wave function
- Wave functions must be continuous, single-valued functions. This condition ensures that the probability density is single-valued and has no discontinuities.
- Wave functions must be normalizable.
- ∇Ψ must be continuous everywhere, except where there is an infinite discontinuity in the potential V. This is necessary since a finite discontinuity in ∇Ψ implies an infinite discontinuity in ΔΨ, and thus from the Schrodinger equation an infinite discontinuity in V.
Define the potential of the one-dimensional harmonic oscillator
The one-dimensional harmonic oscillator potential for angular frequency ω is V(x) = 1/2m(ωx)^2.
Define an even parity state
In one dimension a stationary state wave function satisfying ψ(−x) = ψ(x) is said to describe even parity state.
Define an odd parity state
In one dimension a stationary state wave function satisfying ψ(−x) = -ψ(x) is said to describe odd parity state.
State the allowed energies of a one-dimensional harmonic oscillator
The energies of the one-dimensional quantum harmonic oscillator of angular frequency ω are E = En = (n + ½)ℏω.
Define the complex inner product of a wave function
Let ϕ and ψ be wavefunctions. Then we define the complex inner product ⟨ϕ|ψ⟩ as the integral from -∞ to ∞ of ϕbar(x)ψ(x).
Define a Hilbert space
A complete complex inner product space is called a Hilbert space.
State the Hilbert space postulate
The states of a quantum system are elements of a Hilbert space.
Define adjointness
If A is a linear operator acting on a complex inner product space, then the adjoint A∗ of A satisfies ⟨A∗ϕ|ψ⟩ = ⟨ϕ|Aψ⟩, for all ϕ, ψ ∈ H.
Define self-adjointness
A linear operator A is self-adjoint if A∗ = A.
State the observables postulate
The observables of a quantum system are given by self-adjoint linear operators on the space of states.
Define the position operator
The position operator X in three dimensions has components Xi that are defined on wave functions ψ ∈ H via (Xiψ)(x) = xiψ(x). In vector form, Xψ = xψ.
Define the momentum operator
The momentum operator P has components Pi that are defined on differentiable wave functions ψ by (Piψ)(x) = −iℏ∂ψ/∂(xi)(x). In vector form, Pψ = −iℏ∇ψ.
Define the Hamiltonian operator
The Hamiltonian operator for a particle of mass m moving in a potential V is H = |P|^2/2m + V(X).
Give the Hamiltonian version of the Schrodinger equation
The Schrodinger equation for time-dependent states ψ(t) is iℏ∂ψ(t)/∂t = Hψ(t), with stationary state ψ(t) = ψe^(−iEt/ℏ) of energy E satisfies Hψ = Eψ.
Define an eigenstate
Given an observable A, then a state ψ ∈ H satisfying Aψ = αψ for some constant α ∈ C, is called an eigenstate of A, with eigenvalue α.
Define a spectrum
The spectrum of an observable A is the set of all eigenvalues of A.
State the Dirac span postulate
An observable A in quantum theory is required to have a complete set of eigenstates. That is, there is a set of orthonormal eigenstates {ψn} of A, such that any ψ ∈ H can be written as a linear combination of ψn.
State the quantum measurement postulate
The possible outcomes of a measurement of an observable A are given by the eigenvalues {αn} of A. If the system is in a normalized quantum state ψ, then the probability of obtaining the value αn in a measurement, for a non-degenerate eigenvalue αn, is Pψ(obtaining αn) = |⟨ψn|ψ⟩|2.
State the collapse of the wave function postulate
If one measures the observable A and obtains the non-degenerate eigenvalue αn, then immediately after the measurement the quantum state of the system is the eigenstate ψn ∈ H with Aψn = αnψn.
Define the expectation of an operator
The expectation value of an operator A in a normalized quantum state ψ is Eψ(A) ≡ ⟨ψ|Aψ⟩.
Define the identity operator
The identity operator 1 satisfies 1ψ = ψ, for all ψ ∈ H
Define a non-negative operator
An operator A is said to be non-negative if ⟨ψ|Aψ⟩ ≥ 0, for all ψ ∈ H.
Define the dispersion of an operator
The dispersion of an observable A in a normalized quantum state ψ is ∆ψ(A) ≡ sqrt(Eψ((A − Eψ(A)1)^2) = sqrt(Eψ(A^2) − (Eψ(A))^2).
Define the commutator of two operators
The commutator of two operators A and B is [A, B] ≡ AB − BA.
Give the canonical commutations
[Xi , Pj] = iℏδij1
[Xi, Xj] = 0 = [Pi , Pj].
Give Leibniz’s rule of commutations
[A, BC] = B[A, C] + [A, B]C
Give Jacobi’s identity of commutations
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0.
State Heisenberg’s uncertainty principle
For normalized ψ we have ∆ψ(X)∆ψ(P) ≥ 1/2ℏ and equality holds if and only if ψ(x) = exp[(x − µ)^2/2s0ℏ + γ], for some negative constant s0, and complex constants µ, γ ∈ C.
Define a minimum uncertainty state
A wave function ψ such that ∆ψ(X)∆ψ(P) = 1/2ℏ.
Define the raising/lowering operators
a± = (∓iP + mωX)/sqrt(2mωℏ)
Define the number operator
N = (a+)(a-)
Define the orbital angular momentum operator
The angular momentum operator L = (X2P3 − X3P2, X3P1 − X1P3, X1P2 − X2P1).
Define the Levi-Cevita alternating symbol
The Levi-Civita alternating symbol ϵijk, defined by ϵijk = +1 if ijk is an even permutation of 123, −1 if ijk is an odd permutation of 123 and 0 otherwise.
Define the angular momentum operator
Any self-adjoint vector operator J, with components Ji, i = 1, 2, 3, satisfying [Ji , Jj ] = iℏ times the sum by k of ϵijkJk.is called an angular momentum operator.
Define total angular momentum
The total angular momentum is defined by J^2 = (J1)^2 + (J2)^2 + (J3)^2.
Define the raising and lowering operators of angular momentum
J± = J1 ± iJ2.
Give the Pauli matrices
The 2x2 matrices σ1 = (0 1), (1 0), σ2 = (0 −i) (i 0), σ3 = (1 0) (0 −1).
Give the spin representation of angular momentum
Ji = ℏσi/2, where σi are the Pauli matrices are the matrix representation of spin.
Give Coulemb’s potential
V(r) = e1e2/(4πrϵ0)
Give the Laplacian in spherical coordinates
∇^2 = (1/r^2)d/dr[(r^2)d/dr] + (1/(r^2)sinθ)d/dθ[(sinθ)d/dθ] + (1/(r^2)sin^2(θ))d2/dϕ^2
Give the Bohr radius
The Bohr radius is a = 4πϵ0ℏ^2/me^2.
