13 Harmonic oscillator revisited: Dirac s approach and introduction to Second Quantization

Transcription

1 3 Haronic oscillator revisited: Dirac s approach and introduction to Second Quantization. Dirac cae up with a ore elegant way to solve the haronic oscillator proble. We will now study this approach. The reason we want to study this approach is because this, in fact, gives an alternative approach to quantu echanics and this is known as second quantization. The whole field of quantu field theory is a generalization of the concepts introduced by Dirac in the late 920s. Second quantization also fors an iportant portion of odern day research in quantu theory since atheatics generally becoes sipler in second quantization notation. 2. We define a new operatoraas a linear cobination of the position and oentu operators: a = ı ( ) 2 hω /2ˆp ık/2ˆx (3.) The dual space analogue of this operator is given by: a = ı 2 hω ( ) /2ˆp+ık/2ˆx (3.2) Therefore, a a = [ = hω ı 2 hω ( ) ][ ( /2ˆp+ık/2ˆx ı 2 hω ˆp kˆx2 + ı 2 ) ] /2ˆp ık/2ˆx ( ) /2 k [ˆx, ˆp] (3.3) ˆp Now, [ˆx, ˆp] = ı h, kˆx2 = H, the haronic oscillator Hailtonian and we defined earlier that k = ω 2 as the relation between the force constant and the angular velocity. Therefore, a a = H hω 2 (3.4) Siilarly, [ ( ) aa ı ][ ( ) = 2 hω /2ˆp ık/2ˆx ] ı 2 hω /2ˆp+ık/2ˆx = ˆp2 hω kˆx2 ı ( ) /2 k [ˆx, ˆp] 2 = H hω + 2 (3.5) Cheistry, Indiana University 67 c 2003, Srinivasan S. Iyengar (instructor)

2 Therefore the haronic oscillator Hailtonian can be written as H = 2 hω[ a a+aa ] (3.6) and the coutator [ a,a ] = aa a a = (3.7) 3. Lets also derive a couple of ore coutators[a,h] and [ a,h ] that will be useful later. [a,h] = 2 hω[ aa a+aaa a aa aa a ] = 2 hω[( aa a a ) a+a ( aa a a )] = hωa (3.8) Siilarly, [ a,h ] = 2 hω[ a a a+a aa a aa aa a ] = 2 hω[ a ( a a aa ) + ( a a aa ) a ] = hωa (3.9) 4. Now, let E be an eigenstate ofh with eigenvalue E. Therefore, H E = E E (3.0) Let a E = Q, another ket vector. Therefore, E a a E = Q Q 0 (3.) This iplies E a a E = E H hω 2 E = Ē hω 2 0 (3.2) Therefore, E 2 hω!!! This eans the energy of the quantu Haronic oscillator cannot be lower than 2 hω, hence that is the zero-point energy of the oscillator. (Recall that we got the exact sae result last tie but by solving the differential equation. This tie we have done none of that, we have introduced a new algebraic technique, and we have used that to obtain this result in a uch easier fashion.) Now it reains to see what the operators a and a really ean. 5. For this lets consider the following: ah E = Ea E (3.3) Cheistry, Indiana University 68 c 2003, Srinivasan S. Iyengar (instructor)

3 Using the coutation relations: (Ha+ hωa) E = Ea E (H + hω)[a E ] = E[a E ] H[a E ] = (E hω)[a E ] (3.4) Therefore the ket vector[a E ] is an eigenket ofh with eigenvalue(e hω). 6. Siilarly consider And using the coutation relations: a H E = Ea E (3.5) ( Ha hωa ) E = Ea E (H hω) [ a E ] = E [ a E ] H [ a E ] = (E + hω) [ a E ] (3.6) and the ket vector [ a E ] is an eigenket ofh with eigenvalue(e + hω). 7. We obtained the relatione 2 hω as a result of Eq. (3.2). This eans that the energy can have values only greater than (but including) 2 hω. Lets say: H E 0 = 2 hω E 0 (3.7) where we represent the ket vector corresponding to the energy 2 hω by E Now, using Eq. (3.6) since 2 hω is the energy eigenvalue for the ket E 0. H [ a E 0 ] = ( 2 hω + hω ) [a E 0 ] (3.8) 9. Lets now denote the new eigenvector: E = a E 0. Then we can use Eq. (3.6) again on E to obtain: H [ a E ] = ( 2 hω +2 hω ) [a E ] (3.9) and we could call the new eigenvector: E 2 = a E. We could keep doing this and we will obtain the following for then-th ket vector: H E n = ( 2 hω +n hω ) E n (3.20) Cheistry, Indiana University 69 c 2003, Srinivasan S. Iyengar (instructor)

4 which leads us to the haronic oscillator energy expression: E = ( n+ 2) hω (which we got earlier by solving soe tedious differential equation. This new approach that we have introduced involves a lot less ath (as copared to the differential equation we solved to obtain the Herite polynoials) and we obtain the sae result. 0. However, what does a do when it acts on the eigenket E 0? Lets see if we can answer this question. Lets assue: a E n = β n E n (3.2) where theβ n are to be deterined. This equation also eans: E n a = E n β n (3.22) where we have written the dual space analogue of Eq. (3.2). Fro Eq. (3.2) and Eq. (3.22) we obtain: E n a a E n = β n 2 E n E n = β n 2 (3.23) Using Eq. (3.4) we see that E n a a E n = E n H hω 2 E n = n+ 2 2 = n (3.24) (For this reason a a is called the nuber operator.) Therefore β n = n and a E n = n E n (3.25) This leads to a E 0 = 0. That is the operator a annihilates the state E 0. For this reason a is also called the annihilation operator. For siilar reasons the operator a is called the creation operator. Creation and annihilation operators are extreely iportant in quantu cheistry, since any advanced techniques to solve the tie independent Schrödinger Equation are based on the use of these techniques. The interested reader should look at the following two references for further reading in this subject: (a) H. C. Longuet-Higgins, in Quantu Theory of Atos and Molecules, A TRIBUTE TO JOHN C. SLATER, Ed. P.-O- Löwdin, Acadeic Press 966. p. 05. (b) Sions and Jorgensen, Second Quantization ethods in quantu cheistry.. Now the question: and what is α n. We follow the sae approach as before: a E n = α n E n+ (3.26) E n a = E n+ α n (3.27) and ultiplying the two equations: E n aa E n = α n 2 E n+ E n+ = α n 2 (3.28) Cheistry, Indiana University 70 c 2003, Srinivasan S. Iyengar (instructor)

5 Now using Eq. (3.5) we have E n aa E n = n+ and thereforeα n = n+. Therefore, a E n = n+ E n+ (3.29) 2. Hoework: Use Eqs. (3.29) and (3.25) in Eq. (3.6) to confir the eigenvalues and eigenvectors of the Haronic oscillator. 3. In the next few pages we will see an iportant application of the haronic oscillator proble, that is infra-red spectroscopy. In particular we will see that second quantization can be used to obtain selection rules for IR spectra with little effort. 4. One very iportant reason for studying the haronic oscillator proble is vibrational spectroscopy. To a first approxiation one could assue that the cheical bond is haronic. That is when perturbed fro its equilibriu position the bond tends to relax back to its original equilibriu position. As a result we could assue that a carbon-carbon bond (for exaple) has a spring that connects the two carbon atos as seen in the figure below. (Note this approxiation is only true at low energies. As we approach the dissociation liit the potential energy in the bond deviates substantially fro the haronic potential, as seen in the figure below.) 5. Hence for lower vibrational states, the haronic potential should be a valid approxiation and what we have derived in the previous section could be used to get the properties that one ight see in vibrational spectroscopy. Lets see what we can do here. 6. We have seen before that electroagnetic radiation is a set of perpendicular electric and agnetic field vectors. The frequency of oscillation of these field vectors is proportional to the aount of energy. 7. Can this energy fro the photon be absorbed by a given olecule? Which transitions (otions of the olecule) absorb energy, and which ones dont? 8. Consider a diatoic olecule: Cheistry, Indiana University 7 c 2003, Srinivasan S. Iyengar (instructor)

6 9. Only when a given transition gives rise to a change in dipole oent, will the corresponding frequency be absorbed. 20. Hence the change in dipole oent with respect to a given transition is significant here. 2. In IR spectroscopy, radiation of a certain frequency is incident on the syste, and response is studied and this is what leads to the spectru of the olecule. In ost cases, when the applied radiation is weak, one quantity that is very useful to calculate is the transition dipole bracket (which we will define below), since the response of the syste is proportional to the transition dipole bracket. What this essentially eans is the probability of the transition fro state to state n in the presence of an electroagnetic field (or light) is given by the quantity: P(t) ˆx n 2 (3.30) Eq. (3.30) will not be derived in this class. It can be derived using tie-dependent perturbation theory which we ay not have tie to cover. The interested reader ay look at Chapter 2 of Fayer. 22. However, we note the following. The incident radiation coprises electric and agnetic fields. (We saw earlier, during the SG experients, that light consists of utually orthogonal electric and agnetic field vectors.) The electric field vector interacts with the dipole operator. In fact, the quantity ˆx n is called the transition dipole bracket. Why? It has the units of a dipole (position ties a charge density). And that is why its called the transition dipole bracket. The agnitude of this eleents tells us the probability of transition between states and n in the presence of an external field. 23. Now the probability of transition fro state to state n in the presence of radiation is proportional to the dipole bracket. Lets see if we can evaluate this for the haronic oscillator and get soe results for IR spectroscopy. 24. Using the definition of the creation and annihilation operators in Eqs. (3.) and (3.2) we can write the position operator as: Therefore, ˆx = hω ( a +a ) (3.3) P(t) ˆx n 2 = hω ( a +a ) n 2 = hω ( a +a ) ( n n a +a ) = hω [ a a n n + a n n a + a a ] n n a + a n n (3.32) Cheistry, Indiana University 72 c 2003, Srinivasan S. Iyengar (instructor)

7 Now since [a = ] and [ a = + + ], the first and second ter in the last equation above ust be zero. Therefore: P(t) hω = hω = hω [ a n n a + a n n a ] [ a n 2 + a n 2] [ (n+) n+ 2 +n n 2] (3.33) which can only be non-zero when = n+ or = n. 25. This eans the transition in IR spectroscopy is allowed only between eigenvalues that differ by, if the haronic approxiation is valid. This a vibrational spectru selection rule. Now we have seen earlier that the haronic approxiation is valid for the lower vibrational states, but not for the higher vibrational states (close to dissociation). Hence these selection rules are not valid at higher vibrational states. However, it turns out in practice that even at higher states the ajor contribution does coe fro the lines that differ by one quanta!! 26. Hoework: Using the approach due to Dirac (ie the creation and annihilation operators), derive expressions for x, x 2, p and p 2 for the haronic oscillator. Use this to obtain the uncertainty product x p. Coent on your result. 27. Hoework on Frank-Condon Factors: By solving the electronic Schrödinger Equation, ground and excited electronic states can be obtained at various nuclear configuration like the figure presented in class. In addition, the figure also shows the vibrational levels. Solving for vibrational levels using the nuclear Schrödinger Equation after solving the electronic Schrödinger Equation can get very coplicated but for the sallest of systes. Due to this reason, the vibrational states are assued to exist on a Haronic oscillator potential located at the iniu of the corresponding electronic state as in the figure below. The force constant for the haronic oscillator assues a value such that the curvature of the original surface is identical at the botto of the potential (as you can see fro the figure). This is only an approxiation as you can see, and will work well for the low-lying vibrational levels of each electronic state. While doing optical spectroscopy what happens is the electrons are excited fro the ground electronic state to the excited electronic state as represented by the arrow in the figure. Clearly the vibrational level in the excited electronic state where the electron gets into is iportant. Here s where the Frank-Condon Factors coe in. These represent the probability of such transitions. The transition fro the i-th vibrational level in the ground electronic state to the j-th vibrational level in the first excited electronic state is given by the inner product of the vibrational states ψ0 i ψ j, where ψ i 0 represents thei-th vibrational level in Cheistry, Indiana University 73 c 2003, Srinivasan S. Iyengar (instructor)

8 the ground electronic state, and ψ j represents the j-th vibrational level in the first excited electronic state. (a) Using the definition of the creation operator and the for of the lowest eigenstate of the Haronic oscillator: { [ ]} k exp 2 h (x x 0) 2, (3.34) show that then-th vibrational state of a Haronic oscillator is: ψ0 n ( = ) ] a n k0 exp [ n! 2 h (x x 0) 2 = ( ) ( ω0 n/2 x h ) n exp[ n! 2 h ω 0 x ] k0 2 h (x x 0) 2 (3.35) x 0 is the equilibriu distance and k 0 = ω 2 0 is the force constant of the Haronic oscillator. Note: Using the creation operator will significantly reduce your effort. (b) Using Eq. (3.35) write down an expression for the Frank-Condon factor ψ0 i ψ j, assuing that the equilibriu positions and force constants are different in the ground and excited states, that is {x 0,ω 0,k 0 } for the ground electronic state and {x,ω,k } for the excited electronic state. You do not need to go through the whole integration, just write down the starting expression and explain in words how you would intergrate it, if you were asked to integrate it by hand. (c) Coent on how your derivation would siplify (greatly) if the equilibriu positions and force constants were the sae in the ground and excited states. Cheistry, Indiana University 74 c 2003, Srinivasan S. Iyengar (instructor)