Orbital angular momentum

Problem:

Consider a spinless particle of mass m.  What values can the magnitude of the angular momentum J take in quantum mechanics when it is measured?

Solution:

• Concepts:
  Orbital angular momentum in QM
• Reasoning:
  The only type of angular momentum for this particle is orbital angular momentum. 
• Details of the calculation:
  When measuring L² we only can obtain an eigenvalue l(l +1)ħ², with l a non-negative integer.
  The magnitude of the angular momentum J therefore can only be measured as
  J = (l(l +1))^½ħ, l = 0, 1, 2, ... .

Problem:

The magnitude of the orbital angular momentum L of a hydrogen atom is found to be 30^½ħ.  L_z is measured and found to be 3ħ.
(a)  Which values of the principle quantum number n are consistent with these measurements?
(b)  What is the value of L_x² + L_y²?

Solution:

• Concepts:
  Angular momentum, the hydrogen atom
• Reasoning:
  For the hydrogen atom H, L² and L_z are commuting observables.
• Details of the calculation:
  (a)  L² = l(l+1)ħ² = 30ħ²,  l = 5.  For the hydrogen atom the allowed value for l are all non-negative integers equal to or less than n - 1.  So n must be a positive integer greater or equal to 6.
  (b)  L_x² + L_y² = L² - L_z² = (30 - 9)ħ²  = 21ħ². 

Problem:

A given normalized wave function is ψ(θ,φ) = N sinθ cosφ.
(a)  Write this wave function in terms of spherical harmonics.
(b)  Find the normalization constant N.
(c)  What is the mean value of L² for a state with this wave function?
(d)  L_z is measured.  What are the possible results of this measurement and what are the respective probabilities?

Solution:

• Concepts:
  The spherical harmonics, the postulates of quantum mechanics
• Reasoning:
  If we want to find the probability of measuring the eigenvalue a of an observable A we must express the wave function of the system as a linear combination of eigenfunctions of A.  Then  P(a) = Σ_i|<aⁱ|ψ>|².
• Details of the calculation:
  Express the wave function in terms of spherical harmonics.
  (a)  cosφ = ½(exp(iφ) + exp(-iφ)).
  ψ(θ,φ)  = -(N/2)(8π/3)^½Y₁₁ + (N/2)(8π/3)^½Y_1-1.
  (b)  ∫₀^πsinθ dθ∫₀^2πdφ Y^*_l'm'(θ,φ)Y_lm(θ,φ) = δ_l'lδ_m'm.
  1 = (N²/2)(8π/3).  N² = 2/(8π/3)^½.
  ψ(θ,φ)  = 2^½Y_1-1 - 2^-½/Y₁₁.
  (c)  The given wave function is an eigenfuction of L² with eigenvalue 2ħ².
  (d)  The possible results of measuring L_z are  ħ and -ħ.  The probability of obtaining each of these results is ½.

Problem:

Show that a state with position wave function (y - iz)^k is an eigenstate of the orbital angular momentum operator L_x and find its eigenvalue.  Are there conditions on k?

Solution:

• Concepts:
  Eigenfunctions, orbital angular momentum operators
• Reasoning:
  The orbital angular momentum operator in Cartesian coordinate has the form L = r × p. 
  We check if (y - iz)^k is an eigenfunction of L_x = yp_z - zp_y = (ħ/i)(y∂/∂z - z∂/∂y).
• Details of the calculation:
  ∂(y - iz)^k/∂z = -k*(y - iz)^k-1*i, ∂(y - iz)^k/∂y = k*(y - iz)^k-1.
  L_x(y - iz)^k = (ħ/i)(-iky - kz) (y - iz)^k-1 = -ħk(y - iz)^k.  The eigenvalue is -ħk.
  k must be an integer.

Problem:

For a simple particle moving in space, show that the wave function ψ_lm(r) = x² + y² - 2z² represents a simultaneous eigenstate of L² and L_z with eigenvalues l(l + 1)ħ² and mħ.  Determine l and m.  Find a function with the same eigenvalue for L² and the maximum possible eigenvalue for L_z.

Solution:

• Concepts:
  The eigenfunctions of the orbital angular momentum operator, the spherical harmonics
• Reasoning:
  The common eigenfunctions of L² and L_z are the spherical harmonics.  We have to write the given wave functions in terms of the spherical harmonics.

• Details of the calculation:
  ψ_lm(r) = x² + y² - 2z² = r²sin²θcos²φ + r²sin²θsin²φ - 2r²cos²θ = r²(sin²θ - 2cos²θ)
  in spherical coordinates.
  Express ψ_lm(r) in terms of the spherical harmonics.
  ψ_lm(r) =  r²(1 - cos²θ -  2cos²θ)  = r²(1 - 3cos²θ)
  = -r²(16π/5)^½Y₂₀(θ,φ).
  [Formulas for some of the spherical harmonics:
  Y₀₀ = (4π)^-½,  Y_1±1 = ∓(3/8π)^½sinθ exp(±iφ),  Y₁₀ = (¾π)^½cosθ,
  Y_2±2 = (15/32π)^½sin²θ exp(±i2φ),  Y_2±1 = ∓(15/8π)^½sinθ cosθ exp(±iφ), 
  Y₂₀ = (5/16π)^½(3cos²θ - 1).]
  ψ_lm(r) is an eigenfunction of L_z with eigenvalue 0 and an eigenfunction of L² with eigenvalue  6ħ² (l = 2).  For l = 2 the maximum possible eigenvalue of L_z is mħ = 2ħ.  The corresponding eigenfunction is
  Cr²Y₂₂(θ,φ) = Cr²(15/32π)^½sin²θ exp(±i2φ).

Problem:

The wave function of a particle subjected to a spherically symmetric potential U(r) is given by ψ(r) = (x + y - 3z)f(r).
(a)  Is ψ an eigenfunction of L²?  If so, what is its l value?  If not, what are the possible values of l we may obtain when L² is measured?
(b)  What are the probabilities for the particle to be found in various m_l states?  
(c)  Suppose it is known somehow that ψ(r) is an energy eigenfunction with eigenvalue E.  Indicate how we may find U(r).

Solution:

• Concepts:
  The eigenfunctions of the orbital angular momentum operator, the spherical harmonics
• Reasoning:
  The common eigenfunctions of L² and L_z are the spherical harmonics.  We have to write the given wave functions in terms of the spherical harmonics.
• Details of the calculation:
  ψ(r) = (x + y - 3z)f(r) = (rsinθcosφ + rsinθsinφ - 3rcosθ)f(r)
  = rf(r)(sinθcosφ + sinθsinφ - 3cosθ)
  = rf(r)(sinθ ½(e^iφ + e^-iφ) - i sinθ ½(e^iφ - e^-iφ) - 3cosθ)
  = rf(r)(sinθ ½e^iφ (1 - i) + sinθ ½ e^-iφ (1 + i) - 3cosθ)
  = rf(r)(2^-½exp(-iπ/2) sinθe^iφ  + 2^-½exp(iπ/2) sinθ e^-iφ - 3cosθ).
  
  Y_1±1 = ∓(3/8π)^½sinθ exp(±iφ),  Y₁₀ = (¾π)^½cosθ.
  ψ(r) = rf(r)(-exp(-iπ/2)(4π/3)^½Y₁₁(θ,φ) + exp(iπ/2)(4π/3)^½Y_1-1(θ,φ) - (12π)^½Y₁₀(θ,φ))
  = rf(r)2√π(-exp(-iπ/2)(1/3)^½Y₁₁(θ,φ) + exp(iπ/2)(1/3)^½Y_1-1(θ,φ) - (3π)^½Y₁₀(θ,φ))
  =  (44π/3)^½rf(r)(-exp(-iπ/2)/11^½Y₁₁(θ,φ) + exp(iπ/2)/11^½Y_1-1(θ,φ) - 3/11^½Y₁₀(θ,φ))
  = R(r) (-exp(-iπ/2)/11^½Y₁₁(θ,φ) + exp(iπ/2)/11^½Y_1-1(θ,φ) - 3/11^½Y₁₀(θ,φ)).
  The function of angle is normalized.
  ψ(r) is  an eigenfunction of L² with eigenvalue 2ħ², l = 1.
  
  (b)  The possible values of m_l are 1, 0, -1.
  P(m_l = 1) = 1/11,  P(m_l = -1) = 1/11,  P(m_l = 0) = 9/11.
  
  (c)  The wave function ψ_klm(r,θ,φ) = R_kl(r)Y_lm(θ,φ) = [u_kl(r)/r]Y_lm(θ,φ) is a product of a radial function R_kl(r) and the spherical harmonic Y_lm(θ,φ).  The differential equation for u_kl(r) is
  [-(ħ²/(2m))(∂²/∂r²) + ħ²l(l+1)/(2mr²) + U(r)]u_kl(r) = E_klu_kl(r).
  Since u_kl(r), E_kl and l are known, we can solve this equation for U(r).

Problem:

The ladder operator L₊ is given by
L₊ = ħ exp(iφ) [∂/∂θ + i cotθ ∂/∂φ].
The spherical harmonics are given by Y_lm(θ,φ) = F_lm(θ)exp(imφ).
(a)  Show that Y_ll(θ,φ) = c_l sin^l(θ)exp(ilφ), where c_l is a constant.
(b)  Indicate how you would proceed to find the θ-dependence of Y_lm(θ,φ).

Solution:

• Concepts:
  The raising and lowering operators L₊ = L_x + iL_y  and L_- = L_x - iL_y, the general properties of angular momentum operators
• Reasoning:
  L_±|k,l,m> = [l(l+1) - m(m±1)]^½ħ|k,l,m±1>.  Since
  L_zY_lm(θ,φ) = mħY_lm(θ,φ),  (ħ/i)(∂/∂φ)Y_lm(θ,φ) = mħY_lm(θ,φ),
  we separate variables and look for solutions of the form Y_lm(θ,φ) = F_lm(θ)exp(imφ).
  Wave functions must be continuous at all points in space.
  We therefore need Y_lm(θ,0) = Y_lm(θ,2π) or exp(im2π) = 1 or m = integer.
  Since L₊Y_ll(θ,φ) = 0, we can find a differential equation satisfied F_ll(θ) and solve it.

• Details of the calculation:
  (a)  L₊Y_ll(θ,φ) = ħ exp(iφ) [∂/∂θ + i cotθ ∂/∂φ]Y_ll(θ,φ) = 0
  from the general theory.
  [d/dθ + l cotθ]F_ll(θ) = 0.
  dF_ll(θ) = l cotθ dθ  F_ll(θ), dF_ll(θ) = l F_ll(θ) d(sinθ)/sinθ.
  dF_ll(θ)/F_ll(θ) = l d(sinθ)/sinθ = l d(ln(sinθ)) = d(ln(sinθ))^l = d(sinθ)^l/(sinθ)^l
  -->  F_ll(θ) = c_l(sinθ)^l.
  Y_ll(θ,φ) = c_l sin^l(θ)exp(ilφ).  Y_ll is uniquely defined up to the arbitrary constant c_l.
  ∫₀^πsinθ dθ∫₀^2πdφ |Y_lm(θ,φ)|² = |cl|²∫₀^πsinθ dθ∫₀^2πdφ sin^2l(θ) = 1
  determines |c_l| = (1/(2^ll!))((2l+1)!/(4π))^½.
  It is customary to choose the phase of c_l such that
  |c_l| = [(-1)^l/(2^ll!)] ((2l+1)!/(4π))^½.
  Y_ll(θ,φ) is thus uniquely determined.

  (b) L_-|k,l,m> = [l(l+1) - m(m±1)]^½ħ|k,l,m-1> from the general theory. 
  L_-^l-m|k,l,l> ∝ |k,l,m>.
  L_- is a differential operator.  Apply it (l - m) times to Y_ll(θ,φ) to obtain an expression for Y_lm(θ,φ).  The Y_lm(θ,φ) are thus uniquely defined. 
  To each pair (l,m) there corresponds one and only one eigenfunction Y_lm(θ,φ).

Problem:

A particle in a spherical potential is known to be in an eigenstate of L² and L_z with eigenvalues l(l + 1)ħ² and mħ, respectively.  Prove that the expectation values between |l m> states  satisfy
<L_x> = <L_y> = 0,   <L_x²> = <L_y²> = (l(l + 1)ħ² - m²ħ²)/2.

Solution:

• Concepts:
  The raising and lowering operators L₊ = L_x + iL_y and L₊ = L_x - iL_y,  
  L _±|k,l,m> |k,l,m ± 1> follows from the commutation relations defining angular momentum.
• Reasoning:
  We can express L_x and L_y in terms of L₊ and L_- and then find the matrix elements in the common eigenbasis of L² and L_z.
• Details of the calculation:
  L_x = ½(L₊ + L_-) and L_y = (-i/2)(L₊ - L_-).
  L _±|k,l,m> |k,l,m ± 1>,  <k,l,m|L _±|k,l,m> = 0.
  Therefore <L_x> = <L_y> = 0.
  Eigenstates of L² and L_z have azimuthal symmetry. 
  This symmetry requires that  <L_x²> = <L_y²> = ½<L_x² + L_y²> = ½<L² - L_z²>.
  <L_x²> = <L_y²> = (l(l + 1)ħ² - m²ħ²)/2.

The rigid rotator