**Problem 1:**

Calculate the uncertainty product <Δx><Δp> for the ground state of the infinite square-well potential. How can we tell if the ground-state wave function is or is not a minimum uncertainty state?

Solution:

Concepts: Fundamental assumptions of QM, the eigenstates of the 1D infinite well | |

Reasoning: The eigenfunctions of the infinite well are ψ _{n}(x) = (2/L)^{½}sin(nπx/L), E_{n} = n^{2}π^{2}ħ^{2}/(2mL^{2})
for a well of width L.For any operator A we have ∆A = (<(A - <A>) ^{2}>)^{½} = (<A^{2}>
- <A>^{2})^{½}. | |

Details of the calculation: ∆x = (<x ^{2}> - <x>^{2})^{½}. <x> = L/2 from symmetry.<x ^{2}> = (2/L)∫_{0}^{L} dx x^{2} sin^{2}(πx/L)
= (2L^{2}/π^{3})∫_{0}^{π} dy y^{2} sin^{2}(y)
= (2L ^{2}/π^{3})(π^{3}/6 - π/4) = L^{2}/3 - L^{2}/(2π^{2}).∆x = (L ^{2}/3 - L^{2}/(2π^{2}) - L^{2}/4)^{½}
= (L/2)(⅓ - 2/π^{2})^{½}.∆p = (<p ^{2}> - <p>^{2})^{½}.<p> = 0 for any stationary state. <p ^{2}> = -(2ħ^{2}/L)∫_{0}^{L} dx sin(πx/L) (∂^{2}/∂x^{2})
sin(πx/L)= (2ħ ^{2}π/L^{2})∫_{0}^{π} dy sin^{2}(y)
= (ħ^{2}π^{2}/L^{2}).∆p = (ħπ/L). ∆x∆p = (ħ/2)π(⅓ - 2/π ^{2})^{½} =1.136*(ħ/2).For a minimum uncertainty state, we should have ∆x∆p = (ħ/2) and since 1.136 > 1 we see that the ground state is NOT a minimum uncertainty state. |

**Problem 2:**

Consider a spinless particle of mass m moving in one dimension in the
presence of a delta-function potential well, U(x) = -λ δ(x), λ > 0.

(a) Evaluate the transmission coefficient T(E) as a function of the incident
energy E > 0.

(b) Find the energy of the bound state for this potential well.

(c) Comment on the pole in the expression for T(E) from part (a) in light of
your result

from part (b).

Solution:

Concepts: Piecewise constant potentials | |

Reasoning: This is a "square potential" problem. We solve HΦ(x) = EΦ(x) in regions where U(x) is constant and apply boundary conditions. U(x) = 0 everywhere except at x = 0. | |

Details of the calculation: (a) E > 0. Φ _{1}(x) = A_{1} exp(ikx) + A_{1}'exp(-ikx) for x < 0.
k^{2} = 2mE/ħ^{2}.Φ _{2}(x) = A_{2} exp(ikx) for x > 0.Φ is continuous at x = 0. Φ _{1}(0) = Φ_{2}(0). A_{1
}+
A_{1}' = A_{2}.∂Φ/∂x has a finite discontinuity at x = 0. ∂ ^{2}Φ(x)/∂x^{2} + (2m(E - U)/ħ^{2})Φ(x) = 0.Let us evaluate this equation at x = ε and at x = -ε and write down a difference equation. ∂Φ _{2ε}(x_{1 }+ ε)/∂x - ∂Φ_{1ε}(x_{1
}- ε)/∂x =
-(2m/ħ^{2})∫_{x1-ε}^{x1+ε }(E + λ δ(x)) Φ(x) dx= -(2mλ/ħ ^{2})Φ(0).iA _{1}k – iA_{1}'k = iA_{2}k + A_{2}2mλ/ħ^{2},
A_{1} – A_{1}' = A_{2} + A_{2}2mλ/(ikħ^{2})
= [1 + 2mλ/(ikħ^{2})]A_{2}.Eliminate A _{1}':
2A _{1} = [2 + 2mλ/(ikħ^{2}) ]A_{2}, A_{2}/A_{1}
= 1/[1 + mλ/(ikħ^{2})] = ikħ^{2}/(ikħ^{2} + mλ).T(E) = (k|A _{2}|^{2})/(k|A_{1}|^{2}) = ħ^{4}k^{2}/(ħ^{4}k^{2}
+ m^{2}λ^{2}) = E/(E + mλ^{2}/(2ħ^{2})).(b) E < 0. Φ _{1}(x) = A_{1} exp(ρx) + A_{1}'exp(-ρx) for x < 0. ρ^{2}
= -2mE/ħ^{2}.Φ _{2}(x) = A_{2} exp(ρx) + A_{2}'exp(-ρx) for x > 0.Φ is finite at infinity. A _{1}' = A_{2
}= 0. Φ is continuous
at x = 0. Φ_{1}(0) = Φ_{2}(0). A_{1
}= A_{2}'.∂Φ _{2}/∂x|_{x=0} = ∂Φ_{1}/∂x|_{x=0} - (2mλ/ħ^{2})Φ(0).-ρA _{2}' + (2mλ/ħ^{2})A_{2}' = ρA_{2}', ρ = mλ/ħ^{2}.m ^{2}λ^{2}/ħ^{4} = -2mE/ħ^{2}, E = -mλ^{2}/(2ħ^{2}).Only one bound state exists. (c) The expression for T(E) has a pole when E = -mλ ^{2}/(2ħ^{2})
= energy of the bound state.T --> ∞, but E is negative, we have no incident wave. |

**Problem 3:**

Use what you know about the eigenfunctions of the harmonic oscillator in coordinate space to find the momentum-space eigenfunctions of a harmonic oscillator

Solution:

Concepts: The eigenfunctions of the harmonic oscillator | |

Reasoning: The same equations have the same solutions. | |

Details of the calculation: H = p ^{2}/(2m) +
½mω^{2}x^{2}.In coordinate space (∂ ^{2}/∂x^{2})Φ(x)
+ (2m/ħ^{2})(E
- ½mω^{2}x^{2} )Φ(x)
= 0is the eigenvalue equation for H. The eigenvalues are E _{n}
= (n + ½)ħω,
n = 0, 1, 2, ... . With y = βx and β = (mω/ħ) ^{½}
the eigenvalue equation becomes(∂ ^{2}/∂y^{2})Φ(y)
+ (γ^{2}E
- y^{2} )Φ(y)
= 0, with γ
= (2/ωħ)^{½}.and the normalized eigenfunctions are Φ _{n}(x)
= (n!2^{n})^{-½}(β^{½}/π^{¼})H_{n}(βx)exp(-(βx)^{2}/2),Φ _{n}(y) =
(n!2^{n})^{-½}(1/π^{¼})H_{n}(y)exp(-y^{2}/2).(Note: ∫ _{-∞}^{∞}|Φ_{n}(y)|^{2}dy
= ∫_{-∞}^{∞}|Φ_{n}(y)|^{2}d(βx)
= ∫_{-∞}^{∞}|Φ_{n}(x)|^{2}dx
=1.)Write out the eigenvalue equation for H in momentum space. This is accomplished by replacing x with iħd/dp. [p ^{2}/(2m) -
½mω^{2}ħ^{2}d^{2}/dp^{2}]Φ(p)
= EΦ(p).(d ^{2}/dp^{2})Φ(p)
+ (2/mω^{2}ħ^{2})(E
- p^{2}/(2m))Φ(p)
= 0.With y = αp and α = (mωħ) ^{-½}
the eigenvalue equation becomes(∂ ^{2}/∂y^{2})Φ(y)
+ (γ^{2}E
- y^{2} )Φ(y)
= 0,and the normalized eigenfunctions are Φ _{n}(y) =
(n!2^{n})^{-½}(1/π^{¼})H_{n}(y)exp(-y^{2}/2).Φ _{n}(p)
= (n!2^{n})^{-½}(α^{½}/π^{¼})H_{n}(αp)exp(-(αp)^{2}/2).The eigenvalues are E _{n}
= (n + ½)ħω,
independent of the representation. |

**Problem 4:**

Consider a simple, small, but macroscopic LC circuit made from conventional superconducting material and kept at a temperature below the critical temperature (~1 K). The circuit has no resistance.

(a) Write down a second order differential equation describing the time
evolution of the magnetic flux Φ = LI in the inductor.

(Consider this “the equation of motion” of the circuit.) Relabel C = m, k =
1/L, and compare this equation with the equation of motion of a simple harmonic
oscillator.

(b) Write down a Lagrangian for the LC circuit. (Lagrange’s equation then is
the “equation of motion” of the circuit.) Find the generalized momentum
corresponding to the generalized coordinate in the Lagrangian, and write down
the Hamiltonian for the LC circuit.

(c) Assume that the circuit is cooled down to a
temperature of near zero K (~1 mK). At such a low temperature, only the
lowest allowed energy states are accessible to the system and observable
quantum-mechanical effects can appear. Quantize the system and find the
ground state energy of the system,

(d) Qualitatively reason why excited states of such a system would be unstable
even at very low temperature.

Solution:

Concepts: LC circuits, the Lagrangian and Hamiltonian of a system, quantization of a classical Hamiltonian | |

Reasoning: The same equations have the same solutions. We explore the mathematical analogy between an LC circuit and a harmonic oscillator. | |

Details of the calculation: (a) Φ = LI. emf = -dΦ/dt = Q/C. (Kirchhoff's rule) d ^{2}Φ/dt^{2} + I/C = d^{2}Φ/dt^{2}
+ Φ/(LC) = 0.With C = m and k = 1/L we have d ^{2}Φ/dt^{2} + (k/m)Φ = 0. This is the equation of motion of a simple harmonic oscillator with Φ being a generalized coordinate. (b) L _{Lag} = ½C(dΦ/dt)^{2} - ½Φ^{2}/L.P _{Φ} = ∂L_{Lag}/∂(dΦ/dt) = C(dΦ/dt).H = P _{Φ}^{2}/(2C) + ½Φ^{2}/L.(c) Let P _{Φ} and Φ^{
}become non-commuting operators, [Φ, P_{Φ}]
= iħ.Then H becomes the Hamiltonian operator -ħ ^{2}/(2C) ∂^{2}/∂Φ^{2} +
½Φ^{2}/L,and Hψ(Φ) = Eψ(Φ). The eigenvalues of H = (n + ½)ħω, with ω = 1/(LC) ^{½}. The ground
state energy is ½ħω/(LC)^{½}.(d) We are assuming zero resistance. If kT << ħω/(LC) the probability of thermal excitation and de-excitation is small. But in an excited state, the system radiates. |

**Problem 5:**

Consider a one-dimensional crystal with primitive lattice translation a.

Let
{|n>} be a set orthonormal electron states, n = -∞ to +∞.

Assume that in the
subspace spanned by {|n>} the matrix elements of the electron Hamiltonian are
given by

<n|H|n> = E_{0}, <n|H|_{ }n±1> = -A, <n|H|_{ }n±2> =
B, <n|H|m> = 0 for all other m, with

E_{0} = (7/8)ħ^{2}/(ma^{2}), A = ½ħ^{2}/(ma^{2}),
and B = (1/16)ħ^{2}/(ma^{2}).

(a) Assume the eigenstates of H are of the form |Φ> = ∑_{n} b(x_{n})|n>.

Write down the coupled linear equations for the b(x_{n}). Note that b(x_{n±1})
= b(x_{n} ± a).

(b) Try solutions of the form b(x_{n}) = exp(ikx_{n}) and show
that E as a function of k is given by

E(k) = [ħ^{2}/(ma^{2})][7/8 – cos(ka) + (1/8) cos(2ka)].

(c) Determine the effective mass at the bottom of the energy band and at the
top of the band from a quadratic expansion of E in the departure of k from these
points.

Solution:

Concepts: Electrons in a periodic structure | |

Reasoning: We must solve coupled eigenvalue equations. | |

Details of the calculation: (a) <m|H|Φ> = ∑ _{n} b(x_{n}) <m|H|n> .E b(x _{m}) = E_{0
}b(x_{m}) – A b(x_{m+1}) – A b(x_{m-1}) + B b(x_{m+2}) + B b(x_{m-2})= E _{0} b(x_{m}) – A b(x_{m} + a) – A b(x_{m} -
a) + B b(x_{m} + 2a) + B b(x_{m} – 2a).(b) Eexp(ikx _{n}) = E_{0}expik(x_{n}) – Aexp(ik(x_{n}
+ a))
– Aexp(ik(x_{n} - a)) + Bexp(ik(x _{n} + 2a)) + Bexp(ik(x_{n}
- 2a)).E = E _{0} – Aexp(ika) – Aexp(-ika)) + Bexp(i2ka) + Bexp(i2ka) = E_{0}
– 2Acos(ka) + 2Bcos(2ka).E(k) = [ħ ^{2}/(ma^{2})][7/8 – cos(ka) + (1/8) cos(2ka)].We only need to consider -π/a ≤ k < π/a because k values differing by 2π/a give the same coefficients. The possible eigenvalues of H lie in a band from E _{0}
– 2(A - B) to E_{0} + 2(A – B).(c) An expansion of E(k) about k = 0 yields E(k) = E _{0} - 2(A
- B) + Ak^{2}a^{2} - 4Bk^{2}a^{2}
= E_{min} + Ak^{2}a^{2} - 4Bk^{2}a^{2}.Let E(k) – E _{min} = Ak^{2}a^{2} - 4Bk^{2}a^{2} = ½m*v_{g}^{2}.v _{g} = dω/dk|_{k0} = (1/ħ)dE/dk|_{k0} = (2Aa/ħ)sin(k_{0}a)
- (4Ba/ħ)sin(2k_{0}a).An expansion of v _{g}(k) about k = 0 yields (2Aka^{2}/ħ) - (8Bka^{2}/ħ)
= (ħk/m)½.(ħ ^{2}k^{2}/4m)) = ½ m*(( ħk/(2m))^{2}, m* = 2m.We can also use m* = ħ ^{2}(d^{2}E/dk^{2})^{-1}.
dE/dk = (ħ^{2}k/(2m)) d^{2}E/dk^{2} = (ħ^{2}/(2m)),
m* = 2m.And expansion about k = π/a yields E(k) = E _{max} - Ak^{2}a^{2}
+ 4Bk^{2}a^{2} = E_{max} - (ħ^{2}k^{2}/(4m)).m* = ħ ^{2}(d^{2}E/dk^{2})^{-1} = -2m.Note dv _{g}/dt = F_{ext}/m*. If an external force is
applied, the electron in a crystal is subject to this force and internal
forces from the lattice. The net force may point opposite to the external
force.Remarks: The potential energy function for the one-dimensional crystal is periodic, U(x + a) = U(x). Therefore the Hamiltonian H commutes with the translation operator U(T _{a}).In coordinate representation U(T _{a})f(x) = f(x - a) for any
function f(x).All non-degenerate eigenfunctions of H are eigenfunctions of the translation operator U(T _{a}).Wave functions of the form ψ(x) = e ^{ikx
}v(x) are eigenfunctions of
U(T_{a}) if v(x + a) = v(x), i.e. if v(x) is a periodic
function with period a.U(T _{a}ψ(x) = U(T_{a})e^{ikx
}v(x) = e^{ik(x-a)
}v(x - a) = e^{-ika}e^{ikx }v(x) = e^{-ika}ψ(x).
The eigenvalue is e^{-ika}ψ(x).We can therefore find a complete set of eigenfunctions of H of this form. These functions are called Bloch functions. In coordinate representation |Φ> = ∑ _{n} exp(ikx_{n})|n>
becomes Φ(x) = ∑_{n} exp(ikx_{n})Φ_{n}(x).If we assume that the Φ _{n}(x) = Φ_{0}(x - na), then Φ(x) is
a Bloch function.Φ(x) = ∑ _{n}exp(ikna)Φ_{n}(x) = e^{ikx}∑_{n}exp(-ik(x
- na))Φ_{0}(x - na) = e^{ikx }v(x). v(x) = ∑ _{n}exp(-ik(x - na))Φ_{0}(x - na), n = -∞ to
+∞. v(x + a) = ∑ _{n}exp(-ik(x - (n - 1)a))Φ_{0}(x - (n - 1)a) =
∑_{m}exp(-ik(x - ma))Φ_{0}(x - ma) = v(x).The set {|n>} of orthonormal electron states, n = -∞ to +∞ is not complete. But if we restrict ourselves to this subspace, we may be able to approximately solve for the possible E(k) in one energy band of the crystal. |