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Rotational states of a diatomic molecule can be approximated by those of a rigid rotor. The hamil...
1) Assuming that a diatomic molecule can be approximated by a rigid rotor with a inertia...
1) Assuming that a diatomic molecule can be approximated by a rigid rotor with a inertia momen- tum I = 10–38g cm², calculate the rotational frequency of the radiation that will cause a transition from the J = 1 state to the J = 2 state. In which region of the electromagnetic spectrum this transition will be found?
Consider a rigid heteronuclear diatomic molecule of bond length d rotating in the xy plane about...
Consider a rigid heteronuclear diatomic molecule of bond length d rotating in the xy plane about the z-axis. We have seen that the Hamiltonian for this system is H = P2/(2), where y is the reduced mass, and p = ( y) is the relative momentum. Define the angular momentum operator about the z-axis as L. = fpy - yp... a. Show that Ê, Î2] = 0. b. Show, therefore, that the eigenfunctions of A are also eigenfunctions of L....
3. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the...
3. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the moment of inertia of the rotator. Its rotation is described by a wave function: (0, N{Yo0(0,6)(1 3i) Y1-1(0,6) 2 Y21(0.0) Y20(0.) Find the normalization constant, N. (i) Find the probability to occupy state Yo0- (ii Find the expectation value of L2 of this state (iii Find the expectation value of L2 of this state (iv) Find (L2L2/21...
A. Derive an expression for the rotational partition function in the "high-temperature" limit where qrot can...
A. Derive an expression for the rotational partition function in the "high-temperature" limit where qrot can be approximated as an integral. Remember that the rotational energies as a function of rotational quantum number j are given by: ϵ (j) = B j (j + 1) where B is called the “rotational constant” B = ℏ2 /2µ r 2 , and the degeneracy of each "j" state is D(j) = 2j + 1. B. What is the average rotational energy in...
1. Ideal gas with internal degrees of freedom. Consider a free gas of diatomic molecules at...
1. Ideal gas with internal degrees of freedom. Consider a free gas of diatomic molecules at temperature 7. Diatomic molecules have internal rotational excitations. The rotational energy levels of a single molecule are given by J(J+1) 2/2 J = 0,1,23 where J is the angular momentum and I is the moment of inertia. The degeneracy of the level J is 2J +1. Neglect any interaction between the molecules in the gas. The temperature is high enough so that the statistic...
5. A diatomic molecule (like H2) can be modeled as two atoms of equal mass m,...
5. A diatomic molecule (like H2) can be modeled as two atoms of equal mass m, connected by a rigid massless rod of length a. The system is free to rotate in 3-D. I claim the moment of inertia of this molecule around its ceater of mass is a. (Feel free to convince yourself that factor of k is coect!) Big hint if you 're having trouble getting started: this problem is directly related to McIntyre's Ch A) The energy...
1- 5. Two particles each of mass m are fixed at the end of a rigid...
1- 5. Two particles each of mass m are fixed at the end of a rigid rod of length 2a. This rod lies in the xy plane and is free to rotate in that plane about an axis passing through the midpoint of the rod and perpendicular to it (that is, parallel to the z-axis). Neglect the inertial properties of the rod in the rest of this question z-axis 1. Derive the classical expression for the kinetic energy of the...
1- 5. Two particles each of mass m are fixed at the end of a rigid...
1- 5. Two particles each of mass m are fixed at the end of a rigid rod of length 2a. This rod lies in the xy plane and is free to rotate in that plane about an axis passing through the midpoint of the rod and perpendicular to it (that is, parallel to the z-axis). Neglect the inertial properties of the rod in the rest of this question z-axis 1. Derive the classical expression for the kinetic energy of the...
Heres example 10.2 (3) (30 points) In Example 10.2, the moment of inertia tensor for a...
Heres example 10.2
(3) (30 points) In Example 10.2, the moment of inertia tensor for a uniform solid cube of mass Mand side a is calculated for rotation about a corner of the cube. It also worked out the angular momentum of the cube when rotated about the x-axis - see Equation 10.51. (a) Find the total kinetic energy of the cube when rotated about the x-axis. (b) Example 10.4 finds the principal axes of this cube. It shows that...
1) Assuming that a diatomic molecule can be approximated by a rigid rotor with a inertia momen- tum I = 10–38g cm², calculate the rotational frequency of the radiation that will cause a transition from the J = 1 state to the J = 2 state. In which region of the electromagnetic spectrum this transition will be found?
Consider a rigid heteronuclear diatomic molecule of bond length d rotating in the xy plane about the z-axis. We have seen that the Hamiltonian for this system is H = P2/(2), where y is the reduced mass, and p = ( y) is the relative momentum. Define the angular momentum operator about the z-axis as L. = fpy - yp... a. Show that Ê, Î2] = 0. b. Show, therefore, that the eigenfunctions of A are also eigenfunctions of L....
3. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the moment of inertia of the rotator. Its rotation is described by a wave function: (0, N{Yo0(0,6)(1 3i) Y1-1(0,6) 2 Y21(0.0) Y20(0.) Find the normalization constant, N. (i) Find the probability to occupy state Yo0- (ii Find the expectation value of L2 of this state (iii Find the expectation value of L2 of this state (iv) Find (L2L2/21...
1. Ideal gas with internal degrees of freedom. Consider a free gas of diatomic molecules at temperature 7. Diatomic molecules have internal rotational excitations. The rotational energy levels of a single molecule are given by J(J+1) 2/2 J = 0,1,23 where J is the angular momentum and I is the moment of inertia. The degeneracy of the level J is 2J +1. Neglect any interaction between the molecules in the gas. The temperature is high enough so that the statistic...
5. A diatomic molecule (like H2) can be modeled as two atoms of equal mass m, connected by a rigid massless rod of length a. The system is free to rotate in 3-D. I claim the moment of inertia of this molecule around its ceater of mass is a. (Feel free to convince yourself that factor of k is coect!) Big hint if you 're having trouble getting started: this problem is directly related to McIntyre's Ch A) The energy...
1- 5. Two particles each of mass m are fixed at the end of a rigid rod of length 2a. This rod lies in the xy plane and is free to rotate in that plane about an axis passing through the midpoint of the rod and perpendicular to it (that is, parallel to the z-axis). Neglect the inertial properties of the rod in the rest of this question z-axis 1. Derive the classical expression for the kinetic energy of the...
1- 5. Two particles each of mass m are fixed at the end of a rigid rod of length 2a. This rod lies in the xy plane and is free to rotate in that plane about an axis passing through the midpoint of the rod and perpendicular to it (that is, parallel to the z-axis). Neglect the inertial properties of the rod in the rest of this question z-axis 1. Derive the classical expression for the kinetic energy of the...
Heres example 10.2
(3) (30 points) In Example 10.2, the moment of inertia tensor for a uniform solid cube of mass Mand side a is calculated for rotation about a corner of the cube. It also worked out the angular momentum of the cube when rotated about the x-axis - see Equation 10.51. (a) Find the total kinetic energy of the cube when rotated about the x-axis. (b) Example 10.4 finds the principal axes of this cube. It shows that...
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