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Problem I. A.

a† and a Matrices 1/2

v 1 a† v  v 1 . Sketch the structure of the a† matrix below:

 0

Now sketch the a matrix on a similar diagram.

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Now apply a† to the column vector that corresponds to v = 3.

D. Is a† Hermitian?

E. Is (a† + a) Hermitian? If it is, demonstrate it by the relationship between matrix elements that is the definition of a Hermitian operator.

F . Is i(a† – a) Hermitian? If it is, use a matrix element relationship similar to what you used for part E.

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II.

The Road to Quantum Beats

Consider the 3-level H matrix

Label the eigen-energies and eigen-functions according to the dominant basis state character. The 1!0 state is the one dominated by the zero-order state with E(0) = 10, 0! by E(0) = 0, and – 1!0 by E(0) = –10. A. Use non-degenerate perturbation theory to derive the energies [HINT: H(0) is diagonal, H(1) is non-diagonal]: (i)

E!  10

(ii)

E0! 

(iii)



10

Use non-degenerate perturbation theory to derive the eigenfunctions [HINT: do not normalize]

(i)

1!0  liveexamhelper.com

(ii)

(iii)

0! 

1!0 

Demonstrate the approximate relationship:   H dx  E1!0 [HINT: normalize by dividing by   * !  ! dx .]1!0 1!0

Use the results from part B to write the elements of the T† matrix that non-degenerate perturbation theory promises will give a nearly diagonal H!  T †HT matrix [do not normalize, and do not compute T†HT].

correct elements of the T matrix to write (x, 0) as a linear combination of the eigenstates, 1!0 ,0" , and 1!0 †

[HINT: the columns of T are the rows of T†.]:

For the (x, 0)  c1!01!0  c0"0"  c1!01!0 initial state you derived in part E, write (x, t) (do not normalize). If you do not believe your derived c1!0 , c0! , and c1!0 constants, leave them as symbols.

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Suppose you do an experiment that samples (x,t) by detecting (0 )

fluorescence exclusively form the zero-order  (0 ) 0 0 (x,t). This would be obtained from

character in 2

P (t )   (x,t ) dx P0(t) will be modulated at several frequencies. (i)

(ii)

What is the value of P0(0)?

The contribution of the zero-order 

state to the

observed fluorescence will be modulated at some easily predicted (0 ) frequencies. What are these frequencies? 0

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III.

Inter-Mode Anharmonicity in a Triatomic Molecule

Consider a nonlinear triatomic molecule. There are three vibrational normal modes, as specific in H(0) and two anharmonic inter-mode interaction terms, as specified in H(1).

H (0 )

 1  1 / 2    !N2  ! 1 N 2 122 Q 1 Q 2  k 2233 2 32 2

H(1)  k

3 1 3 /2  1 / 2    !N



Q Q

List all of the (∆v1, ∆v2, ∆v3) combined selection rules for nonzero matrix elements of the k term122in H(1)? One of these selection rules is (+1, +2, 0).

List all of the (∆v1, ∆v2, ∆v3) selection rules for nonzero matrix (1) elements of the k term 2233 in H ?

In the table below, in the last column, place an X next to the inter- mode vibrational anharmonicity term to which the k2233 term contributes . (i)

(ii)

e12

e23

!z e e2233

 1 / 2  v  1 / 2  1

v

!x e

(iii)

v

!x

 1 / 2  v  1 / 2  2

v

 1 / 2 2

v

 1 / 2

Does the term you specified in part C depend on the sign of k2233?

(1) Does the k122 term in H give rise to any vibrational anharmonicity terms that are sensitive to the sign of k122? Justify your answer. liveexamhelper.com

IV. Your First Encounter with a Non-Rigid Rotor Your goal in this problem is to compute the v-dependence of the rotational constant of a harmonic oscillator. Some equations that you will need: B(R) 

h 4  cµ R2

Q̂  R  R e 1 R



B e

h e 4  cµ R2

† h 1/2   4  cµ ˆ ˆ a a   e

1  Q  Re 

 21 Q Re  R

2

1

Power series expansion:

thus

 1   2   Q 3 2 Q   3 Q 12 1 2  R R  Re   Re   4e  R   , e 

 B(R)  B 1 2 e  Some algebra yields Q   B e1/2    e Re

Q    2 3 Q   Re   Re 

 .  (1)

aˆ  aˆ † 

 Be 

 103 , an excellent order-sorting parameter.   e  1/2   2 e Hˆ ROT hcB J (J e 1)1 2   † e aˆ  aˆ †  B   B    3  aˆ  aˆ    e   e  

where

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(2)

From boxed equation (2), what is Hˆ ( 0 ) ?

What is Hˆ (1) ?

C. What is E ( 0 ) , as a function of hc, B , and J(J + 1)? e J , as a function of hc, B ,  , (v + 1/2), and J(J + 1)?

What is E

J [HINT: (a a + aa†) = (2N + 1).]

†

(1 )

From experiment we measure E

E

E

 hcB J (J 1)

Bv  Be   e  v 1 / 2  ,

Bv1  Bv   e .

What is e expressed in terms of hc, Be, and e?

Does the sign you have determined by e bother you? Why?

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Derivation of One Part of the Angular Momentum Commutation Rule

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Solution I.

a† and a Matrices A.

v 1 a† v   v 11/2 . Sketch the structure of the a† matrix below:

Now sketch the a matrix on a similar diagram.

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Now apply a† to the column vector that corresponds to |v = 3]

Is a† Hermitian?

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Is (a† + a) Hermitian? If it is, demonstrate it by the relationship between a matrix element that is the definition of a Hermitian operator

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Is i(a† – a) Hermitian? If it is, use a matrix element relationship similar to what you used for part E.

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II. The Road to Quantum Beats Consider the 3-level H matrix

Use non-degenerate perturbation theory to derive the energies [HINT: H(0) is diagonal, H(1) is non-diagonal]:

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B. Use non-degenerate perturbation theory to derive the eigenfunctions [HINT: do not normalize]

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Use the results from part B to write the elements of the T† matrix that non-degenerate perturbation theory promises will give a nearly diagonal H! = T† HT matrix [do not normalize].

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If you obtained an answer you believe in part G, you will have discovered quantum beats. Even if you are not convinced that your answer to part G is correct, you will receive partial credit for being as explicit as you can be about P0(t):

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III. Inter-Mode Anharmonicity in a Triatomic Molecule Consider a nonlinear triatomic molecule. There are three vibrational normal modes, as specific in H(0) and two anharmonic inter-mode interaction terms, as specified in H(1)

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In the table below, in the last column, place an X next to the intermode vibrational anharmonicity term to which the k2233 term contributes

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Does the term you specified in part C depend on the sign of k2233?

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IV. Your First Encounter with a Non-Rigid Rotor Your goal in this problem is to compute the v-dependence of the rotational constant of a harmonic oscillator. Some equations that you will need:

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From boxed equation (2), what is Hˆ (0) ?

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D. From experiment we measure

Does the sign of αe bother you? Why? One might expect that as v increases, Bv will decrease. This is correct for an anharmonic non-rigid rotor. However, for a harmonic non-rigid rotor, Bv will increase with v. This occurs because B(R) increases more at the inner turning point than it decreases at the outer turning point.

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Derivation of One Part of the Angular Momentum Commutation Rule

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Some Possibly Useful Constants and Formulas

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