Perturbation theory (Ch. 6) (time-independent, nondegenerate, Sec. - - PowerPoint PPT Presentation

EE201/MSE207 Lecture 15

Perturbation theory (Ch. 6)

(time-independent, nondegenerate, Sec. 6.1)

Usually solving TISE 𝐼𝜔 = 𝐹𝜔 is too complicated; need approximations. Perturbation theory is one of approximate methods to solve TISE.

𝐼 = 𝐼0 + 𝐼1

Idea: separate Hamiltonian into simple and small parts (if possible) where Trick:

𝜔𝑜 = 𝜔𝑜

(0) + 𝜇𝜔𝑜 (1) + 𝜇2𝜔𝑜 (2)+ . . .

𝐼0𝜔 = 𝐹𝜔 is simple (can be solved),

𝐼1 is small

𝐼 = 𝐼0 + 𝜇 𝐼1

then power series in 𝜇 ≪ 1, and then 𝜇 = 1

𝐹𝑜 = 𝐹𝑜

(0) + 𝜇𝐹𝑜 (1) + 𝜇2𝐹𝑜 (2)+ . . .

𝐼0 + 𝜇 𝐼1 (𝜔𝑜

0 + 𝜇𝜔𝑜 1 + . . . ) = (𝐹𝑜 0 + 𝜇𝐹𝑜 1 + . . . )(𝜔𝑜 0 + 𝜇𝜔𝑜 1 + . . . )

• rder 𝜇0:

𝐼0𝜔𝑜

0 = 𝐹𝑜 0 𝜔𝑜

(solvable exactly)

• rder 𝜇1:

𝐼0𝜔𝑜

1 +

𝐼1𝜔𝑜

0 = 𝐹𝑜 0 𝜔𝑜 1 + 𝐹𝑜 1 𝜔𝑜

• rder 𝜇2:
• rder 𝜇3:

. . . . . .

. . .

notation 𝜔 = |𝜔〉

First-order perturbation theory (𝜇1)

notation 𝜔1 𝐼 𝜔2 = 〈𝜔1| 𝐼 𝜔2〉 Multiply by 𝜔𝑜

0 𝑦 ∗

and integrate,

−∞ ∞ 𝑒𝑦, or (the same) 〈𝜔𝑜 0 |. . .

𝐼0𝜔𝑜

1 +

𝐼1𝜔𝑜

0 = 𝐹𝑜 0 𝜔𝑜 1 + 𝐹𝑜 1 𝜔𝑜

〈𝜔𝑜

(0)|

𝐼0|𝜔𝑜

1 〉 + 〈𝜔𝑜 0 |

𝐼1|𝜔𝑜

0 〉 = 𝐹𝑜 0 〈𝜔𝑜 0 |𝜔𝑜 1 〉 + 𝐹𝑜 1

equal

𝐹𝑜

1 = 〈𝜔𝑜 0 |

𝐼1|𝜔𝑜

0 〉

First-order correction to energy is just the average (expectation) value of 𝐼1 in the unperturbed state (very natural)

First-order perturbation theory (cont.)

Now find correction 𝜔𝑜

(1) to wavefunction

𝐼0𝜔𝑜

1 +

𝐼1𝜔𝑜

0 = 𝐹𝑜 0 𝜔𝑜 1 + 𝐹𝑜 1 𝜔𝑜

𝜔𝑜

(1) = 𝑛≠𝑜 𝑑𝑛 (𝑜) 𝜔𝑛 (0)

Multiply by 〈𝜔𝑚

(0)| :

Rewrite

( 𝐼0 − 𝐹𝑜

0 ) 𝜔𝑜 1 = −(

𝐼1 −𝐹𝑜

(1)) 𝜔𝑜

Expand in zero-order eigenstates

𝑑𝑜

(𝑜) = 0

from normalization

𝑛≠𝑜(𝐹𝑛

(0) − 𝐹𝑜 0 ) 𝑑𝑛 (𝑜) 𝜔𝑛 0 = −(

𝐼1 −𝐹𝑜

(1)) 𝜔𝑜

(𝐹𝑚

0 − 𝐹𝑜 0 ) 𝑑𝑚 𝑜 = − 𝜔𝑚

𝐼1 𝜔𝑜 + 𝐹𝑜

1 𝜀𝑚𝑜

For 𝑜 = 𝑚 we obtain the previous formula for 𝐹𝑜

(1) (another way of derivation)

For 𝑜 ≠ 𝑚: 𝑑𝑚

𝑜 = 𝜔𝑚

𝐼1 𝜔𝑜 𝐹𝑜

0 − 𝐹𝑚

Rename 𝑚 → 𝑛

𝜔𝑜

(1) = 𝑛≠𝑜

𝜔𝑛 𝐼1 𝜔𝑜 𝐹𝑜

0 − 𝐹𝑛

𝜔𝑛

(0)

𝐹𝑜 = 𝐹𝑜

(0) + 𝐹𝑜 (1) + …

First-order perturbation theory: summary

• Remark. Correction to 𝜔𝑜 is not good if 𝐹𝑛

(0) = 𝐹𝑜 (0) (i.e. when degeneracy). Then

the formalism is a little different. Usually degeneracy is lifted (disappears) due to

• perturbation. For example, in hydrogen atom there is fine structure (due to

relativistic correction and spin-orbit) and hyperfine structure (due to magnetic interaction of electron and proton).

𝜔𝑜

(1) = 𝑛≠𝑜

𝜔𝑛 𝐼1 𝜔𝑜 𝐹𝑜

0 − 𝐹𝑛

𝜔𝑛

(0)

𝜔𝑜 = 𝜔𝑜

(0) + 𝜔𝑜 (1) + …

𝐹𝑜

1 = 〈𝜔𝑜 0 |

𝐼1|𝜔𝑜

0 〉

𝐼 = 𝐼0 + 𝐼1 𝐼0𝜔𝑜

0 = 𝐹𝑜 0 𝜔𝑜

Second-order perturbation: similar but lengthier

Result for second-order correction to energy of nth level:

𝐹𝑜

(2) = 𝑛≠𝑜

𝜔𝑛 𝐼1 𝜔𝑜

2

𝐹𝑜

0 − 𝐹𝑛

WKB approximation (Ch. 8)

𝐹 can be discrete

• r continuous

Idea:

Two cases: 𝐹 > 𝑊(𝑦) (classical region)

𝐹 < 𝑊(𝑦) (tunneling)

(Wentzel, Kramers, Brillouin, 1926)

− ℏ2 2𝑛 𝑒2𝜔 𝑦 𝑒𝑦2 + 𝑊 𝑦 𝜔 𝑦 = 𝐹 𝜔(𝑦)

If 𝑊 𝑦 = const, then easy to solve If 𝑊 𝑦 varies slowly, then modify solution for 𝑊 𝑦 = const.

WKB approximation, classical region, 𝐹 > 𝑊(𝑦)

If 𝑊 𝑦 = const = 𝑊, then 𝜔 𝑦 = 𝐵 𝑓±𝑗𝑙𝑦,

𝑙 = 2𝑛(𝐹 − 𝑊) ℏ 𝜔 𝑦 ≈ 𝐵 𝑦 exp ±𝑗

𝑦 𝑙 𝑦′ 𝑒𝑦′

𝐾 = 𝑗ℏ 2𝑛 𝜔 𝑒𝜔∗ 𝑒𝑦 − 𝜔∗ 𝑒𝜔 𝑒𝑦

Then for 𝑊(𝑦) we expect slightly different in the textbook, ±

𝑗 ℏ 𝑞 𝑦 𝑒𝑦

From conservation of the probability current we obtain

𝐵 𝑦 ∝ 1 𝑙(𝑦)

Therefore

𝜔 𝑦 ≈ const 𝑙 𝑦 exp ±𝑗

𝑦

𝑙 𝑦′ 𝑒𝑦′ 𝑙(𝑦) = 2𝑛[𝐹 − 𝑊 𝑦 ] ℏ

Remark 1. 1 𝑙(𝑦) ∝ 1 𝑤 𝑦 , so 𝜔 2 ∝ 1 𝑤(𝑦), as it should be. Remark 2. If 𝑛(𝑦) (as in SiGe technology), then 𝐵(𝑦) ∝ 𝑛(𝑦) 𝑙(𝑦). Remark 3. WKB approximation works well only for slowly changing 𝑊(𝑦).

WKB approximation, tunneling, 𝐹 < 𝑊(𝑦)

If 𝑊 𝑦 = const = 𝑊, then 𝜔 𝑦 = 𝐵 𝑓±𝜇𝑦,

𝜇 = 2𝑛(𝑊 − 𝐹) ℏ

Similarly

𝜔 𝑦 ≈ const 𝜇 𝑦 exp ±

𝑦

𝜇 𝑦′ 𝑒𝑦′ 𝜇(𝑦) = 2𝑛[𝑊 𝑦 − 𝐹] ℏ

WKB approximation is often used to estimate probability of tunneling (coefficient of transmission) through a strong (almost “opaque”) tunnel barrier Tunneling probability

𝑈 ≃ 𝜔out 2 𝜔in 2 𝑈 ≃ exp(−2

𝑏 𝜇 𝑦 𝑒𝑦)

𝑈 ⋘ 1

(very crudely; we neglect all pre-exponential factors, which usually are within

• ne order of magnitude, while exponential factor can typically be 10−3 − 10−10)

𝑏 𝜔𝑗𝑜 𝜔𝑝𝑣𝑢 𝐹 𝑊(𝑦)

WKB, connection between the two regions

𝑊(𝑦) 𝐹 𝑦0 (classical turning point)

WKB approximation does not work well in the vicinity of 𝑦0, we need a better approximation near 𝑦0 (linear potential, Airy functions). Result:

𝜔 𝑦 ≈ 2 𝐷 𝑙(𝑦) sin 𝜌 4 +

𝑦 𝑦0

𝑙 𝑦′ 𝑒𝑦′ , 𝑦 < 𝑦0 𝐷 𝜇(𝑦) exp −

𝑦0 𝑦

𝜇 𝑦′ 𝑒𝑦′ , 𝑦 > 𝑦0

assume smooth 𝑊(𝑦) (different result for an abrupt potential)

Variational principle (Ch. 7)

Only Sec. 7.1 Theorem: For an arbitrary |𝜔〉, the ground state energy 𝐹𝑕 satisfies inequality

𝐹𝑕 ≤ 𝜔 𝐼 𝜔 = 〈 𝐼〉

Proof is simple. Let us expand 𝜔 = 𝑜 𝑑𝑜|𝜔𝑜〉. Then since 𝐹𝑜 ≥ 𝐹𝑕, we get

𝐼 = 𝑜 𝑑𝑜 2𝐹𝑜 ≥ 𝐹𝑕 𝑜 𝑑𝑜 2 = 𝐹𝑕

This theorem can be useful to estimate 𝐹𝑕 (or at least to find an upper bound) Idea: Use trial wavefunctions |𝜔〉 with many adjustable parameters and minimize 〈 𝐼〉. Hopefully min 〈 𝐼〉 is close to 𝐹𝑕. Extensions of this method can also be used to find |𝜔𝑕〉, first-excited state energy and wavefunction (using subspace orthogonal to |𝜔𝑕〉), second-excited state, etc.