### Create a StudySoup account

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

Already have a StudySoup account? Login here

# PHYSICAL CHEMISTRY CHEM 455

UW

GPA 3.92

### View Full Document

## 17

## 0

## Popular in Course

## Popular in Chemistry

This 33 page Class Notes was uploaded by Carmela Kilback on Wednesday September 9, 2015. The Class Notes belongs to CHEM 455 at University of Washington taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/192589/chem-455-university-of-washington in Chemistry at University of Washington.

## Similar to CHEM 455 at UW

## Reviews for PHYSICAL CHEMISTRY

### 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: 09/09/15

Using QM on simple systems the free particle Classically describe particle motion by 2 0 Fmam 2 dt Integrate equation to obtain xx0V0t Particle moves with constant velocity Modeling free particle with QM Use Schrodinger Time Independent wave equation h2d2wx m M VltxgtwltxgtEwltxgt Because Vx 0 d 2wx 2m de hZ Solve by inspection with help from before solving Maxwell s Equation QM solutions for wave functions Two linearly independent solutions Ex if x Aeikx Ae and x if xAeikx Ae Where k 2 sz l h2 E l l h zat To get LPxt multiply by e e Solutions are plane waves Rewrite solutions as A e ikx z wt These are plane waves moving to right and left Not eigenfunctions of fc Calculate P nding particle in interval LltxltL wxwxdx AAeikxeikxdx dx PXdx L L ikx z39kx Z Lw xwxdx AA Le 6 dx PX independent of X What does this mean 4 What are the total energy eigenvalues Substitute solutions into Schrodinger equation Both give same answer h2 d2wx h2 d2Aeikx h2k2 A eikx 2m a x2 2m a x2 2m Because there are no restrictions on k the energy spectrum is continuous Are these eigenfunctions of H also eigenfunctions of f9 Only way to nd out is to try ihiAeikx ial hkAeikx ial 8x Are they eigenfunctions and if so What are the eigenvalues for momentum Also notice that H and 3 Commute in this case Compare with X and p not commuting Next con ne particle to 1D box Potential energy de ned by Vx0agtxgt0 Vxoox2axSO Like putting one molecule In 3D box QM for particle in box Schrodinger equation is d2wx 2m dx2 dzw x d to be nite W X 0 outside box and atx0andxa For This is called a boundary condition Use real representation for wx Most general solution is wx AsinkxBcoskx For w0 0 andWa 0 w0 O B 0 Therefore B O wa Asinka O Quantization of k result of boundary condition Revisit wa Asin ka 2 O Only sens1ble ch01ce 1s ka mz k a Eigenfunctions of H are w x Asinmzxj n 1234 a n is a quantum number and k is quantized because of con nement or binding Normalize eigenfunctions Normalization condition is V x mm AAa sin2 0 jalle A 3 a mrx a Normalized total energy eigenfunctions are wakemm Calculate energy eigenvalues Substitute eigenfunctions in Sch equation 2 2 2 2 JLMEX 2 Fsmmj 2 EM x 2m d x 2m 61 a a 2 2 En h n 123 2m 61 The minus sign on the operator made the resulting energy positive Discrete energy spectrum Eigenfunetions EF and quantization View En wnx EFs of time H independent Scum Schr equation g 5 Are standing EN LIJ waves Quantization is the result of con nement w 6 must have nodes at ends of box Therefore 9 X is a standing wave Add 12 wavelength in going to n1 EF Note zero point energy gt 0 What would xi have to be to make zero point energy 0 Note that AE between levels increases as n increases Correspondence principle Probability distribution in box How do you measure position and momentum 2 mm 125 9030012 E in units of h28ma2 What is probability for classical particle Must approach classical result as n gt 00 Higher resolution Lower resolution 1 39 n50 1 n50 1 n50 08 08 08 02 02 02 02 04 06 08 I 02 04 06 08 I 02 04 06 08 I 1 n30 r1 3 n30 l 08 W HM l1 0 ifl I 06 l W ll 11 06 04 02 l l 02 0 I I I 08 08 08 06 06 n21 06 n21 04 04 04 02 02 02 02 04 06 08 I 02 04 06 08 I 02 04 06 08 I L 11111111111111 l J1 8 1 02 0 All 1111 1111111111111 l l 08 02 04 06 08 I l l Example Problem 43 What is the probability P of nding the particle in the central third of the box if it is in its ground state Solution 2 7 x For the ground state W10 5m 7 From the postulate we can see that P is the sum of all the probabilities of nding the particle in intervals of width dx within the 211 central third of the box This probability is given by the integral P 2 j smz jdx Solving this integral as in Example Problem 42 a Z a 3 Although we can39t predict the outcome of a single measurement we can predict that for 61 of a large number of individual measurements the particle will be found in the central third of the box What would the probability be for nding a classical particle in this interval 17 2D particle in box Potential Vxyz O for O lt x lt a 0 lt y lt b 00 otherwise a b c d e f The wave function is a product of the two independent parts and the Energy is the sum 18 w w 90 y y 722 quotMy 2ma2 Count Nodes in X and Y and rank order the energies Example Problem Consider the function lx csin E of sin 27 Cl Cl a Is wx an acceptable wave function for the particle in the box b Is wx an eigenfunction of the total energy operator H C Is wx normalized d How will this wave function evolve in time a Is W X an acceptable wave function for the particle in the box Boundary conditions First and second derivatives well behaved over interval of interest Square integral or normalizable 20 Is wx an eigenfunction of the total energy operator H Let operator act on w x h2 d2 7m Z xD 2 csm d51n 2mdx a a 2 2 1 7r 2 csinfmj 4dsin27mD ma a a 21 Is W06 normalized a x 27rsz csm ds1n a a Evaluate I dx 2 x 2 27rx c csm jd dsm 0 Cl Cl cd cdsinmjsin27rxj Cl Cl 2 J c zsin2 jdx I dlzsin227r xjdx 0 a 0 a ajcd cdsin jsin2 jdx 0 Cl Cl dx 0Q 22 ls w X normalized continued Result is a 2 lcz sin W sin dx 0 Cl Cl C z a asin 27 sin0 2 47 Half 3 W 042 W 7239 2csinmj dsin normalized if xZ a a W W 1 23 An easier way to nd the norm Use the orthonormal properties of our eigenfunctions and write the wave function in terms of them nltxgt sm Icomxmnowxwm f c 1ltxdx 111 myMight xd ltxc 1ltxd 2 em gcc dd E ccdd a 24 How does it evolve in time I X not standing wave 2 L11363972 CSln elwlt 018111 72x e zwzt a 61 En 2 ha Values of observables will in general be time dependent HOW does the energy evolve in time 25 Are the Particle in Box total energy EF also EF of la Test by carrying out operation d I l 39xj ihmzx 2 n x zh 51n 2 cos dx a a a a a Can only determine expectation value lt19 If x wxdx 0 26 p for particle in box total B EF Calculate using 3rd postulate ltpgt amw 2 Isinn xj ihisin mmde a 0 a dx a 21h2mr JSinmrxjcosmrxjdx a a a 0 sin2 mt sin2 0 0 a What about ltXgt 27 Example Problem 44 Assume that a particle is con ned to a box of length a and that the system wave function is 7m lx s1n a a a Is this state an eigenfunction of the position operator b Calculate the average value of the position x that would be obtained for a large number of measurements Explain your result Solution 2 a The position operator x Because xyx x sin i cyx a a b Where c is a constant we conclude that the wave function is not an eigenfunction of the position operator We calculate the expectation value using the fourth postulate 28 a a 2 x sin sin dx ij sin dx a 0 a a a 0 a x2 cos 2bx Using the standard integral Ix sin bx 2dx 4 x Sm 2 0 2721C 2721C a 2 cos xs1n 2 2 2 x a a 2 a a gt7 T 2 7r a a 0 The average position is midway in the box This is exactly what we would expect because the particle is equally likely to be in each half of the box 29 How to do well in Chem 455 Recognize that this is a demanding course budget the time needed Read the material before class Ask questions about what you don t understand Master problem solVing Work all questions and problems in text Start with the ones in red and look at answers if stuck Learn from your mistakes Make mistakes Use course resources office hours Thursday tutorial sample exams homework solutions example problems How do you know whether you know the material Do problems Make up problems Look in other books for problems Look at the problems before you read the text while you read the text after you read the text Look at assigned problems right away Spend under 15 min on each Do not do the problem unless it falls right out Come back to it the next day Classical physics can t explain why an atom with a positive nucleus and surrounding electrons is stable chemical bonds are formed between atoms copper conducts electricity but diamond doesn t light emission spectrum in gaseous discharges occurs only at certain frequencies energy at the atomic level is quantized Device invention based on QM Transistor Laser UV Vis or IR spectrometer NMR and EPR spectrometer MRI which is NMR imaging Arc or uorescent lamp Superconductors Light Emitting Diodes Plasma Screen

### 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

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I used the money I made selling my notes & study guides to pay for spring break in Olympia, Washington...which was Sweet!"

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "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.