Quantum_Mechanics_Flashcards_Part1_2
What equation represents a simple sine wave in quantum mechanics?
Ψ(x, t) = A sin(kx − ωt)
What is the phase velocity of a wave?
v_phase = ω / k
What is the complex form of a wave used in quantum mechanics?
Ψ(x, t) = A exp[i(kx − ωt)]
How is probability related to the wave function in quantum mechanics?
P(x, t) = |Ψ(x, t)|² = Ψ*Ψ
What principle relates localization in space to uncertainty in momentum?
Heisenberg Uncertainty Principle: Δx Δp ≥ ℏ/2
What are the assumptions used to derive the Schrödinger Equation?
- E = ℏω, 2. p = ℏk, 3. Energy conservation E = T + V
What is the time-dependent Schrödinger Equation?
iℏ ∂Ψ/∂t = [−ℏ²/(2m) ∂²/∂x² + V(x,t)]Ψ
What is the time-independent Schrödinger Equation?
−ℏ²/(2m) d²ψ/dx² + V(x)ψ = Eψ
What does the wavefunction Ψ(x, t) represent?
A complete description of a quantum state.
What are the properties of the probability density function P(x, t)?
Real, non-negative, normalized: ∫|Ψ|² dx = 1
How is the expectation value of position defined?
⟨x⟩ = ∫ Ψ* x Ψ dx
What is the momentum operator in 1D?
p̂ = −iℏ ∂/∂x
What is the expectation value of momentum?
⟨p⟩ = ∫ Ψ* (−iℏ ∂/∂x) Ψ dx
What is the kinetic energy operator?
T̂ = −ℏ²/(2m) ∂²/∂x²
What is the Hamiltonian operator?
Ĥ = T̂ + V̂ = −ℏ²/(2m) ∂²/∂x² + V(x)
How is a step potential defined in quantum mechanics?
V(x) = 0 for x ≤ 0, V₀ for x > 0
What are the boundary conditions at a potential step?
Continuity of ψ and dψ/dx at the boundary
What is quantum tunnelling?
Penetration of a particle into a classically forbidden region where E < V
What is the transmission probability through a potential step?
T = 4k₀k / (k₀ + k)²
What is the reflection probability?
R = ((k₀ − k)/(k₀ + k))²
What are the boundary conditions in an infinite square well?
ψ = 0 at the walls, ψ must be continuous inside
What are the energy eigenvalues for an infinite square well?
Eₙ = n²π²ℏ²/(2ma²), n = 1,2,3,…
What does the quantum number n represent?
It labels the allowed energy levels of a bound system
What is zero-point energy?
The lowest possible energy of a quantum system, not zero due to HUP
What is the potential energy in the quantum harmonic oscillator?
V(x) = ½ mω²x²
What are the energy eigenvalues of the quantum harmonic oscillator?
Eₙ = (n + ½)ℏω
What type of functions are the wavefunctions for the quantum harmonic oscillator?
ψₙ(x) = Hₙ(u) e^(−u²/2), where Hₙ are Hermite polynomials
What does the time-dependent wavefunction Ψ(x,t) describe?
The full quantum state, including time evolution.
What is a stationary state?
A state where the probability density |Ψ|² is time-independent.
What causes oscillations in probability density for a superposition state?
Interference between energy eigenstates with different eigenvalues.
What is Postulate 3 in quantum mechanics?
Any wavefunction can be expressed as a linear combination of eigenstates of a complete basis.
What is the Copenhagen interpretation of measurement?
A measurement collapses the wavefunction to a corresponding eigenstate.
What are Hermitian operators?
Operators with real eigenvalues; they represent observable quantities.
What is orthogonality of eigenfunctions?
Eigenfunctions of a Hermitian operator are orthogonal: ∫ψₙ*ψₘ dx = δₙₘ
How is the expectation value of an operator A in a superposition state calculated?
⟨A⟩ = ∑|cₙ|²aₙ, where aₙ are eigenvalues and cₙ are coefficients.
What does it mean for two operators to commute?
Their commutator is zero: [A, B] = 0, so they can share eigenfunctions.
Can non-commuting operators be simultaneously measured?
No, their measurements affect one another.
What is the commutator of position and momentum?
[x, p̂] = iℏ
What is the Heisenberg uncertainty principle?
Δx Δp ≥ ℏ/2 for position and momentum.
Why do non-commuting operators lead to uncertainty?
Because their eigenfunctions are not shared.
How does Fourier analysis relate to uncertainty?
A narrow wavepacket in position has a wide distribution in momentum.
What is the 3D time-independent Schrödinger equation?
−ℏ²/(2m) ∇²ψ + V(r)ψ = Eψ
What is the Laplacian in Cartesian coordinates?
∇² = ∂²/∂x² + ∂²/∂y² + ∂²/∂z²
What is a separable solution for a 3D box?
ψ(x,y,z) = X(x)Y(y)Z(z), where each satisfies a 1D TISE.
What is the energy for a particle in a 3D box?
E = (ℏ²π²/2m)(n₁²/a² + n₂²/b² + n₃²/c²)
What is degeneracy in a 3D box?
Different quantum number combinations result in the same energy.
What is the classical expression for angular momentum?
L = r × p
What are the components of the angular momentum operator?
L̂ₓ = −iℏ(y∂/∂z − z∂/∂y), etc.
Do angular momentum components commute?
No: [L̂ₓ, L̂ᵧ] = iℏL̂𝓏 and cyclic permutations.
What is the operator for total angular momentum squared?
L̂² = L̂ₓ² + L̂ᵧ² + L̂𝓏²
Which components of angular momentum can be known simultaneously?
L² and one component (usually L̂𝓏)
What is the Coulomb potential for hydrogen?
V(r) = −Ze²/(4πε₀r)
What are the three quantum numbers for hydrogen?
n (principal), l (orbital), m (magnetic)
What is the energy of the nth level in hydrogen?
Eₙ = −13.6 Z² / n² eV
What are spherical harmonics?
Yₗᵐ(θ, φ) = N Pₗᵐ(cosθ) e^{imφ}, eigenfunctions of angular momentum operators.
What determines degeneracy in hydrogen?
Energy depends only on n; multiple (l, m) values share the same energy.
