### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# QuantumMechanicsI PHYS516

Drexel

GPA 3.54

### View Full Document

## 35

## 0

## Popular in Course

## Popular in Physics 2

This 2 page Class Notes was uploaded by Vernice Schuster on Wednesday September 23, 2015. The Class Notes belongs to PHYS516 at Drexel University taught by RobertGilmore in Fall. Since its upload, it has received 35 views. For similar materials see /class/212524/phys516-drexel-university in Physics 2 at Drexel University.

## Similar to PHYS516 at Drexel

## Popular in Physics 2

## Reviews for QuantumMechanicsI

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/23/15

TimeIndependent Perturbation Theory Robert Gilmore Physics Department Drezel University Philadelphia Pennsylvania 19104 USA Dated January 30 2007 My Physics Class Quantum Mechanics l Perturbation theory is introduced by diagonalizing a 3 X 3 matrix Generalization to a larger basis is immediate This treatment is simpler than the usual treatment and leads immediately to results one higher order than usual in both the perturbed eigenvalues and eigenfunctions I INTRODUCTION Computing eigenvaluesvectors is all about diagonal izing matrices nite gtlt nite or 00 X 00 We treat the problem Ella 95 95 9 H05H1 6 E2Ib 6 1 x x 9 If some eigenvalues of H0 are degenerate eg the 2p lev els in hydrogen then corresponding submatrices should first be diagonalized for example Ella H1ij where H1ij is the submatrix of 6H1 in the subspace of states that have degenerate eigenvalue E1 of H0 11 A SIMPLE PERTURBATION PROBLEM The method we propose is valid for Hg with nondegen erate eigenvalues All the results we need can be deter mined by using a simple 3 X 3 matrix and then applying the rules of general covariance77 at the end of the calcu lation We want to compute the eigenvalues and eigenvectors of the 3 X 3 matrix H0 6H1 We begin by constructing the secular equation from E1 ehu 7 A eh12 ehis eh21 E2 eh22 7 A eh23 7gt Ehgl 6h32 E3 Shag A E1 ehu 7 A E2 eh22 7 A E3 Ehgg 7 A 63h23h31h12 h21h13h32 62E1 Ehii Ah23h32 62E2 6h22 Ah31h13 62E3 Shag Ah12h21 2 where e H1j 7gt ehij To determine the eigenvalues we set the determinant equal to zero We shall solve for the perturbation of the eigenvalue E2 To do this we set the determinant equal to zero divide by E1 ehu 7 A E3 Ehgg 7 A and rearrange the equation to find E2 eh22 7 A 71462 7 B62 7 063 3 so that AE2eh2252AB530 4 The terms A B C in this expression are E2 6h22 Ah31h13 A E1 ehu 7 A E3 Ehgg 7 A h21h12 123132 B 7 7 5 E1 ehu 7 A E3 Ehgg 7 A C h23h31h12 h21h13h32 E1 ehu 7 A E3 ehgg 7 A The coef cients A B C are functions of 6 Since A 7 E2 eh22 is of order 62 cf Eq 3 the term A62 is fourth order and can be neglected if we wish to compute corrections to E2 only to third order The coef cients B and C have Taylor series in 6 beginning with a constant term To construct the correction to the energy E2 to third order it is suf cient to replace A 7 E2 eh22 in the denominators of B and C We find 3 h 39h39 E3Eh2 2112 2 2 6 22 6 E2Ej6h22ihjj 3 3 h2 39h 391th 63 J J 1 2 E2 EjE2 Ek 6 TABLE 1 Terms of order 6 and 62 obtained by multiply ing out the matrices in Eq 1 Order 1 Index h21111 h23u3 3 u3E3 E2 h32 v3E3 E2 Us H33 h22 III INCREASING THE BASIS The result to third order valid for any Hg with non degenerate levels and arbitrary H1 is simply obtained y 2 A i and removing the limits 3 in the summations above Principle of General Covariance 3 7 H 2 hijhjz39 E 7 E T 6hquot T 6 Ez39 E 60 hjj hijhjkhm l 3 wemwem m k i 1n the familiar Dirac form this is E113 Ei ltil6H1ligt Z lzl6 lllfwlilflw WEWWEW J a uc Ei EjEi Ek In this expression E10 Ej ltjleHlljgt IV WAVEFUNCTIONS Expressions for the perturbed wavefunctions are ob tained by similar methods We rst write down the eigen vector equation for a generic perturbed wavefunction to second order in the smallness parameter 6 eu1 62m 1 eug e2v2 eus 62 X 0 These two matrices are multiplied out Terms of order 6 and e2 are collected in Table 1 The perturbed vector to second order in e is E1 E2 6h11 h22 6h12 Ehis ehgl 7B0e2 ehgg 6h31 6h32 E3 E2 6h33 h22 Order 2 1 u1E1 7 E2 h12 v1E1 7 E2 u10111 h22 h12112 hisus B 0 h31u1 h32u2 h h h 7h h h 0 5 E13132 62 1212311113252Jr E17E122 227E2 1 7 m 2 h32h327h22 hglhlg 0 5 E27E2gt 5 JFTngEN E27E2E17E2 1 From this result we can write down the general result by inspection and substitution vwkmezm i J l 39 15H 16Hllill6H12 mm wrgyll m T m E 7 EMEk 7 El This is the standard result in timeindependent pertur bation theory V CONCLUSION We have simpli ed the presentation of time independent perturbation theory by presenting it for small 3 X 3 matrices then extending in the obvious way to arbitrarily sized matrices ACKNOWLED GMENT S isinterest in the standard presentations of this subject The author thanks several former classes for expressing falling asleep and snoring REFERENCES 1 L E Ballentine Quantum Mechanics A Modern De velopment Singapore World Scienti c 1998

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made $280 on my first study guide!"

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.