A particle is represented (at time t = 0) by the wave function

ψ(x,t) = A(a^{2} - x^{2}) if -a < x < a, ψ(x,t) = 0
otherwise.

(a) Determine the normalization constant A.

(b) What is the expectation value of x (at time t = 0)?

(c) What is the expectation value of p (at time t = 0)?

(d) Find the expectation value of x^{2}.

(e) Find the expectation value of p^{2}.

(f) Find the uncertainty in x (Δx).

(g) Find the uncertainty in p (Δp).

(h) Check that your results are consistent with the uncertainty principle.

Solution:

- Concepts:

Postulates of quantum mechanics - Reasoning:

We use the given wave function to calculate the mean value and rms deviation of observables. - Details of the calculation:

(a) We need A^{2}∫_{-a}^{a}(a^{2}− x^{2})^{2}dx = 2∫_{0}^{a}(a^{2}− x^{2})^{2}dx = A^{2}16a^{5}/15 = 1.

A = √15 a^{-5/2}/4.

(b) <x> = ∫_{-∞}^{∞}|Ψ(x)|^{2}x dx = A^{2}∫_{-a}^{a}(a^{2}- x^{2})^{2}x dx = 0.

|Ψ|^{2}is even, x is odd and the integral is zero.

(c) <p> = (ħ/i)∫_{-∞}^{∞}Ψ*(x) (dΨ(x)/dx) dx = -A^{2}(ħ/i)∫_{-a}^{a}2x (a^{2}- x^{2}) dx = 0.

(d) <x^{2}> = ∫_{-∞}^{∞}|Ψ(x)|^{2}x^{2}dx = A^{2}∫_{-a}^{a}(a^{2}- x^{2})^{2}x^{2}dx = a^{2}/7.

(e) <p^{2}> = -ħ^{2}∫_{-∞}^{∞}Ψ*(x) (d^{2}Ψ(x)/dx^{2}) dx = A^{2}ħ^{2}∫_{-a}^{a}2(a^{2}- x^{2}) dx = (5/2)(ħ^{2}/a^{2}).

(f) Δx = (<x^{2}> - <x>^{2})^{½}= a/√7.

(g) Δp = (<p^{2}> - <p>^{2})^{½}= (5/2)^{½}(ħ/a).

(h) The uncertainty principle is satisfied.

ΔxΔp = (a/√7)(5/2)^{½}(ħ/a) = (5/14)^{½}ħ > ħ/2.

An operator A, representing observable A, has two normalized eigenstates
Ψ_{1} and Ψ_{2},
with eigenvalues a_{1} and a_{2}, respectively. An operator
B, representing observable B, has
two normalized eigenstates Φ_{1} and Φ_{2}, with eigenvalues b_{1}
and b_{2}, respectively. The eigenstates
are related by

Ψ_{1} = (3Φ_{1} + 4Φ_{2})/5,

Ψ_{2} = (4Φ_{1} - 3Φ_{2})/5.

(a) Observable A is measured, and the value a_{1} is obtained. What is the state of
the system immediately after this measurement?

(b) If B is now measured immediately following the A measurement, what are the possible
results, and what is the probability of measuring each result?

(c) Right after the measurement of B, A is measured again. What is the
probability of getting a_{1}?

Solution:

- Concepts:

The postulates of Quantum Mechanics - Reasoning:

When a physical quantity described by the operator A is measured on a system in a normalized state |ψ>, the probability of measuring the eigenvalue a_{n}is given by

P(a_{n}) = Σ_{i=0}^{gn}|<u_{n}^{i}|ψ>|^{2}, where {|u_{n}^{i}>} (i=1,2,...,g_{n}) is an**orthonormal basis**in the eigensubspace E_{n}associated with the eigenvalue a_{n}.

If a measurement on a system in the state |ψ> gives the result a_{n}, then the state of the system immediately after the measurement is the normalized projection of |ψ> onto the eigensubspace associated with a_{n}. - Details of the calculation:

We assume the system has no time to evolve between measurements.

(a) After measuring a_{1}, the system is in the eigenstate ψ_{1}.

(b) If B is now measured, the probability of obtaining b_{1}is 9/25 and the probability of obtaining b_{2}is 16/25.

(c) There are two path. If b_{1}is measured the system is in the state φ_{1}= (3ψ_{1}+ 4ψ_{2})/5.

The probability of measuring a_{1}after measuring b_{1}is 9/25.

If b_{2}is measured the system is in the state φ_{2}= (4ψ_{1}- 3ψ_{2})/5.

The probability of measuring a_{1}after measuring b_{2}is 16/25.

The probability of getting a_{1}again after a measurement of B is (9/25)^{2}+ (16/25)^{2}= 0.5392.

(a) Show that the eigenvalues of a general 2×2 matrix A can be expressed as

λ_{±}= ½T ±
(¼T^{2} - D)^{½}, **
**where D is the determinant of A and T is the trace of A (sum of diagonal
elements).

Show that the matrix

A =

0 | m | ||

m | M |

with M >> m has two eigenvalues, with one much larger than the other.

(b) Show that the most general form of a 2×2 unitary matrix U with unit determinant can be parameterized as

U =

a | b | ||

-b* | a* |

,

subject to the constraint aa^{*} + bb^{* }= 1,
where ^{* }denotes complex conjugation.

Solution:

- Concepts:

Mathematical foundations of quantum mechanics - Reasoning:

This is a linear algebra problem. - Details of the calculation:

(a) To find the eigenvalues of an n×n matrix A requires solving the characteristic (also termed secular) equation det(A - λI) = 0 for λ, with I the n×n unit matrix, For the general 2×2 matrix

A =

a b c d we have

a - λ b c d - λ = 0.

(a - λ)(d - λ) - bc = 0.

λ_{±}= ½(a + d) ± ((a + d)^{2}- 4(ad - bc))^{½}= ½T ± (¼T^{2}- D)^{½},

where T = Tr(A) = a + b and D = det(A) = ad - bc.

Applying this to the matrix

A =

0 m m M

and assuming M >> m gives T = M, D = -m^{2},

λ_{±}= ½M ± (¼M^{2}+ m^{2}) = ½M ± ½M(1 + 4m^{2}/M^{2})^{½}≈ ½M ± ½M(1 + 2m^{2}/M^{2})

= ½M ± (½M + m^{2}/M).

The eigenvalues therefore are λ_{+}≈ M, λ_{-}≈ m^{2}/M, |λ_{+}| >> |λ_{-}|.

(b) Parameterize the matrix in terms of complex parameters a, b, c, and d, and require the unitarity condition UU^{† }= I.U =

a b c d U

^{†}=a* c* b* d* .

UU

^{† }= I --> aa^{*}+ bb^{* }= 1, cc^{*}+ dd^{* }= 1, ac^{*}+ bd^{* }= 0. b = -c^{*}a/d^{*}.

det(U) = 1 --> ad - bc = 1, ad + |c|^{2}a/d^{*}= 1, a(|d|^{2}+ |c|^{2}) = d^{*}, a = d^{*}, b = -c^{*}.

ThereforeU =

a b -b* a*

|a|

^{2}+ |b|^{2}= 1.

The corresponding function of an operator Ω is defined as F(Ω) = ∑

Let |Φ

F(Ω)|Φ

|Φ

Consider a three-state system. In some orthonormal basis, {|1>, |2>, |3>}, the matrix of the Hamiltonian operator H is

H =

0 | 2 | 0 | ||

2 | 0 | 0 | ||

0 | 0 | 1 |

.

Choose units such that ħ = 1. Find the matrix of the evolution operator in the same basis.

Solution:

- Concepts:

Functions of operators, change of basis - Reasoning:

Use the eigenbasis of H to evaluate a function of the operator H. - Details of the calculation:

The eigenvalues of H are E, where

-E 2 0 2 -E 0 0 0 1-E = 0.

E^{2}(1 - E) - 4(1 - E) = 0. The solutions to this equation are E_{1}= -2, E_{2}= 1, E_{3}= 2.

The corresponding normalized eigenvectors are|E

_{1}> = 2^{-½}-1 1 0 , |E

_{2}> =0 0 1 , |E

_{3}> = 2^{-½}1 1 0 .

The matrix of H in its eigenbasis, with the eigenvectors ordered as {|E

_{1}>, |E_{2}>, |E_{3}>}, isH =

-2 0 0 0 1 0 0 0 2 .

The matrix of the evolution operator in the eigenbasis of H is

U' = exp(-iHt) =

exp(i2t) 0 0 0 exp(-it) 0 0 0 exp(-i2t) .

The matrix of the evolution operator in the original {|1>, |2>, |3>} basis is

U= DU'D^{†}, where D is the unitary transformation from the {|1>, |2>, |3>} basis to the {|E_{1}>, |E_{2}>, |E_{3}>} basis.

The matrix of D isD =

-2 ^{-½}0 2 ^{-½}2 ^{-½}0 2 ^{-½}0 1 0 .

The matrix of D

^{†}isD

^{†}=-2 ^{-½}2 ^{-½}0 0 0 1 2 ^{-½}2 ^{-½}0 .

Therefore the matrix of the evolution operator in the original {|1>, |2>, |3>} basis is

U =

cos(2t) -isin(2t) 0 -isin(2t) cos(2t) 0 0 0 exp(-it) .

Consider a two state system with kets |0> and |1> defining a set which is
assumed to be orthonormal and complete.

The Hamiltonian of the system is H = ħωa^{†}a, where the operators a and a^{†}are defined by

a^{†}|0> = |1>, a|1> = |0>, a^{†}|1> = a|0> = 0.

(a) Find the eigenvalues of H.

(b) Write down the Dirac (bra - ket) representation of the operators H, a, and
a^{†}.

(c) What are the anti-commutators of the operators a and a^{†}?

Solution:

- Concepts:

Fundamental assumptions of QM - Reasoning:

We are given enough information to construct the matrix of the Hermitian operator H in the given basis. - Details of the calculation:

(a) H|0> = ħωa^{†}a|0> = 0 --> E_{0}= 0, |0> is an eigenvector of H.

H|1> = ħωa^{†}a|1> = ħω|1> --> E_{1}= ħω, |1> is an eigenvector of H.

(b) For any operator O we have = ∑_{n}∑_{m}|n><n|O|m><m|.

H = 0|0><0| + ħω|1><1| = ħω|1><1|.

a = |0><1|, a^{†}= |1><0|.

(c) aa^{†}= |0><0|, a^{†}a = |1><1|, {a, a^{†}} = aa^{†}+ a^{†}a = I.