Principles of Quantum Mechanics Flashcards
What is the Hamiltonian?
The sum of the kinetic and potential energies of all particles associated with the system.
What is the equation for wavenumber, k?
k = 2π / λ = 2πf / c
State the time-dependent Schrodinger equation for a single particle of mass m in 1 dimension in a potential V (x).
(i ̄h) ∂Ψ / ∂t = - ( ̄h^2) / (2m) * (∂^2Ψ) / (∂x^2) + V(x) Ψ(x,t)
State what it means for a wavefunction |φ〉to be an eigenfunction of an operator ˆA.
|φ〉 is an eigenfunction of an operator ˆA if:
ˆA|φ = λ |φ〉
where λ is the eigenvalue corresponding to the eigenfunction |φ〉.
(State what it means for a wavefunction |φ〉to be an eigenfunction of an operator ˆA.)
Supposing the operator ˆA represents a physical quantity, what is the physical interpretation of the set of eigenvalues of ˆA?
The set of eigenvalues of ˆA corresponds to
the possible outcomes of a measurement
of the physical quantity represented by ˆA.
Give the expression for the energy levels E(nx ny nz) of the 3-dimensional harmonic oscillator.
E(nx, ny, nz) = (nx + ny + nz + 3 / 2) ̄hω
(Give the expression for the energy levels E(nx ny nz) of the 3-dimensional harmonic oscillator.)
What are the allows values of the quantum numbers nx, ny, nz?
Where:
nx = 0, 1, 2, …
ny = 0, 1, 2, …
nz = 0, 1, 2, …
(Give the expression for the energy levels E(nx ny nz) of the 3-dimensional harmonic oscillator.)
(What are the allows values of the quantum numbers nx, ny, nz?)
Give an example of degeneracy for the 3-dimensional harmonic oscillator.
E(100) = E(010) = E(001) = 5 / 2 ̄hω
What is the dirac notation?
〈φ|ψ〉 =∫ (∞, −∞) φ∗ψ dx
How do you find the complex conjugate of something?
You find the complex conjugate by changing the sign of the imaginary part of the complex number.
State the defining property of an even function f(x).
A function f (x) is called even when:
f (-x) = f (x)
State the defining property of an odd function f(x).
A function f (x) is called odd when:
f (-x) = -f (x)
Is the product of an even and an odd function even or odd?
The product of an even and an odd function is odd.
Describe the quantum mechanical phenomenon of tunneling.
Tunneling is a quantum mechanical phenomenon
when a particle is able to penetrate through a potential energy barrier
that is higher in energy than the particle’s kinetic energy.
Describe a technological application of tunneling.
A technological application of the phenomenon of tunneling is the Scanning Tunnelling Microscope (STM).
Here an atomically sharp tip is being moved across a surface.
The tunneling current of electrons (between the tip and the
surface) is measured and from this, the distance between the tip and the surface can be determined.
By scanning the surface, a height profile can be obtained at atomic resolution
Express the z-component L(z) of the classical angular momentum (->)L in terms of the coordinates x, y, z and the momenta p(x), p(y), p(z).
L(z) = x p(y) - y p(x)
Express the x-component L(x) of the classical angular momentum (->)L in terms of the coordinates x, y, z and the momenta p(x), p(y), p(z).
L(x) = y p(z) - z p(y)
Express the y-component L(y) of the classical angular momentum (->)L in terms of the coordinates x, y, z and the momenta p(x), p(y), p(z).
L(y) = z p(x) - x p(z)
What is the quantum-mechanical operator ^L(z)?
^L(z) = x ^p(y) - y^p(x)
Give the equation for the Hamiltonian.
H^ = - (ħ^2 / 2m) (∂^2 / ∂x^2) + V(x)
What is the operator form of the Schrodinger equation?
^H ψ = E ψ
What is the equation for energy levels?
E(n) = (n + 1/2) ħ ω
What is Born’s rule?
P(x) = |ψ(x, t)|^2
What is a “good” quantum number?
If [ ˆA, ˆH ] = 0,
then eigenfunctions of ˆA at an initial point in time remain eigenfunctions for all time.
The corrresponding physical quantity A is conserved.
In this case, the quantum number associated with the operator
ˆA is said to be a good quantum number.
What is the Heisenberg Uncertainty Principle?
∆x ∆p ≥ ħ / 2
Where:
∆x is the uncertainty in position
∆p is the uncertainty in momentum
State the equation of motion of a classical harmonic oscillator with mass m and spring constant k.
m (..)x = F = - k x
ω = √(k / m)
