lecture 16: quantized energies - bucknell universityphys211/sp2020/cal/materials/lect16.pdflecture...

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Lecture 16: Quantized Energies 1. Wavefunctions and Energy Levels 2. Emission & Absorption 3. Classically Allowed & Forbidden Regions 4. Energy Levels for Multiple Atoms - ~ 2 2m d 2 (x) dx 2 + U (x) (x)= E (x) For infinite square well: n (x)= r 2 L sin nL x E n = n 2 h 2 8mL 2 E photon = |E final - E initial | For absorption or emission of a photon: Schrödinger’s Equation:

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Page 1: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption

Lecture 16: Quantized Energies

1.Wavefunctions and Energy Levels

2.Emission & Absorption 3.Classically Allowed &

Forbidden Regions 4.Energy Levels for

Multiple Atoms

� ~22m

d2 (x)

dx2+ U(x) (x) = E (x)

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For infinite square well:

n(x) =

r2

Lsin

⇣n⇡L

x⌘

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En = n2 h2

8mL2<latexit sha1_base64="wZstyrjrnyXaYtdTBUYUp2xKs7g=">AAACBXicbVDLSgMxFM34rPU16lIXwSK4KjNFsBuhKIILFxXsA9rpkEkzbWiSGZKMUIZu3Pgrblwo4tZ/cOffmGlnoa0HLvdwzr0k9wQxo0o7zre1tLyyurZe2Chubm3v7Np7+00VJRKTBo5YJNsBUoRRQRqaakbasSSIB4y0gtFV5rceiFQ0Evd6HBOPo4GgIcVIG8m3j659AS+g6FVgN5QIp8NeZZJW+a1pRd8uOWVnCrhI3JyUQI66b391+xFOOBEaM6RUx3Vi7aVIaooZmRS7iSIxwiM0IB1DBeJEeen0igk8MUofhpE0JTScqr83UsSVGvPATHKkh2rey8T/vE6iw6qXUhEnmgg8eyhMGNQRzCKBfSoJ1mxsCMKSmr9CPEQmDG2Cy0Jw509eJM1K2XXK7t1ZqXaZx1EAh+AYnAIXnIMauAF10AAYPIJn8ArerCfrxXq3PmajS1a+cwD+wPr8AdIdlts=</latexit><latexit sha1_base64="wZstyrjrnyXaYtdTBUYUp2xKs7g=">AAACBXicbVDLSgMxFM34rPU16lIXwSK4KjNFsBuhKIILFxXsA9rpkEkzbWiSGZKMUIZu3Pgrblwo4tZ/cOffmGlnoa0HLvdwzr0k9wQxo0o7zre1tLyyurZe2Chubm3v7Np7+00VJRKTBo5YJNsBUoRRQRqaakbasSSIB4y0gtFV5rceiFQ0Evd6HBOPo4GgIcVIG8m3j659AS+g6FVgN5QIp8NeZZJW+a1pRd8uOWVnCrhI3JyUQI66b391+xFOOBEaM6RUx3Vi7aVIaooZmRS7iSIxwiM0IB1DBeJEeen0igk8MUofhpE0JTScqr83UsSVGvPATHKkh2rey8T/vE6iw6qXUhEnmgg8eyhMGNQRzCKBfSoJ1mxsCMKSmr9CPEQmDG2Cy0Jw509eJM1K2XXK7t1ZqXaZx1EAh+AYnAIXnIMauAF10AAYPIJn8ArerCfrxXq3PmajS1a+cwD+wPr8AdIdlts=</latexit><latexit sha1_base64="wZstyrjrnyXaYtdTBUYUp2xKs7g=">AAACBXicbVDLSgMxFM34rPU16lIXwSK4KjNFsBuhKIILFxXsA9rpkEkzbWiSGZKMUIZu3Pgrblwo4tZ/cOffmGlnoa0HLvdwzr0k9wQxo0o7zre1tLyyurZe2Chubm3v7Np7+00VJRKTBo5YJNsBUoRRQRqaakbasSSIB4y0gtFV5rceiFQ0Evd6HBOPo4GgIcVIG8m3j659AS+g6FVgN5QIp8NeZZJW+a1pRd8uOWVnCrhI3JyUQI66b391+xFOOBEaM6RUx3Vi7aVIaooZmRS7iSIxwiM0IB1DBeJEeen0igk8MUofhpE0JTScqr83UsSVGvPATHKkh2rey8T/vE6iw6qXUhEnmgg8eyhMGNQRzCKBfSoJ1mxsCMKSmr9CPEQmDG2Cy0Jw509eJM1K2XXK7t1ZqXaZx1EAh+AYnAIXnIMauAF10AAYPIJn8ArerCfrxXq3PmajS1a+cwD+wPr8AdIdlts=</latexit><latexit sha1_base64="wZstyrjrnyXaYtdTBUYUp2xKs7g=">AAACBXicbVDLSgMxFM34rPU16lIXwSK4KjNFsBuhKIILFxXsA9rpkEkzbWiSGZKMUIZu3Pgrblwo4tZ/cOffmGlnoa0HLvdwzr0k9wQxo0o7zre1tLyyurZe2Chubm3v7Np7+00VJRKTBo5YJNsBUoRRQRqaakbasSSIB4y0gtFV5rceiFQ0Evd6HBOPo4GgIcVIG8m3j659AS+g6FVgN5QIp8NeZZJW+a1pRd8uOWVnCrhI3JyUQI66b391+xFOOBEaM6RUx3Vi7aVIaooZmRS7iSIxwiM0IB1DBeJEeen0igk8MUofhpE0JTScqr83UsSVGvPATHKkh2rey8T/vE6iw6qXUhEnmgg8eyhMGNQRzCKBfSoJ1mxsCMKSmr9CPEQmDG2Cy0Jw509eJM1K2XXK7t1ZqXaZx1EAh+AYnAIXnIMauAF10AAYPIJn8ArerCfrxXq3PmajS1a+cwD+wPr8AdIdlts=</latexit>

Ephoton = |Efinal � Einitial|<latexit sha1_base64="caFM+jlPy1xIx45vy/zHYtRI7u8=">AAACJXicbZDLSgMxFIYzXmu9jbp0EyyCG8uMCLpQKIrgsoK9QFtKJs20oZlkSM4Ipe3LuPFV3LiwiODKVzHTFqqtBwI/3zmHk/8PYsENeN6Xs7S8srq2ntnIbm5t7+y6e/tloxJNWYkqoXQ1IIYJLlkJOAhWjTUjUSBYJejepv3KE9OGK/kIvZg1ItKWPOSUgEVN9+quWY8IdHTUjzsKlBziazyYwZBLIob4FM8Qlxy4hYNs0815eW9ceFH4U5FD0yo23VG9pWgSMQlUEGNqvhdDo080cCrYMFtPDIsJ7ZI2q1kpScRMoz92OcTHlrRwqLR9EvCY/t7ok8iYXhTYyfSnZr6Xwv96tQTCy4a1FSfAJJ0cChOBQeE0MtzimlEQPSsI1dY8xbRDNKFgg01D8OctL4ryWd738v7Dea5wM40jgw7RETpBPrpABXSPiqiEKHpGr+gdjZwX5835cD4no0vOdOcA/Snn+we6t6X+</latexit><latexit sha1_base64="caFM+jlPy1xIx45vy/zHYtRI7u8=">AAACJXicbZDLSgMxFIYzXmu9jbp0EyyCG8uMCLpQKIrgsoK9QFtKJs20oZlkSM4Ipe3LuPFV3LiwiODKVzHTFqqtBwI/3zmHk/8PYsENeN6Xs7S8srq2ntnIbm5t7+y6e/tloxJNWYkqoXQ1IIYJLlkJOAhWjTUjUSBYJejepv3KE9OGK/kIvZg1ItKWPOSUgEVN9+quWY8IdHTUjzsKlBziazyYwZBLIob4FM8Qlxy4hYNs0815eW9ceFH4U5FD0yo23VG9pWgSMQlUEGNqvhdDo080cCrYMFtPDIsJ7ZI2q1kpScRMoz92OcTHlrRwqLR9EvCY/t7ok8iYXhTYyfSnZr6Xwv96tQTCy4a1FSfAJJ0cChOBQeE0MtzimlEQPSsI1dY8xbRDNKFgg01D8OctL4ryWd738v7Dea5wM40jgw7RETpBPrpABXSPiqiEKHpGr+gdjZwX5835cD4no0vOdOcA/Snn+we6t6X+</latexit><latexit sha1_base64="caFM+jlPy1xIx45vy/zHYtRI7u8=">AAACJXicbZDLSgMxFIYzXmu9jbp0EyyCG8uMCLpQKIrgsoK9QFtKJs20oZlkSM4Ipe3LuPFV3LiwiODKVzHTFqqtBwI/3zmHk/8PYsENeN6Xs7S8srq2ntnIbm5t7+y6e/tloxJNWYkqoXQ1IIYJLlkJOAhWjTUjUSBYJejepv3KE9OGK/kIvZg1ItKWPOSUgEVN9+quWY8IdHTUjzsKlBziazyYwZBLIob4FM8Qlxy4hYNs0815eW9ceFH4U5FD0yo23VG9pWgSMQlUEGNqvhdDo080cCrYMFtPDIsJ7ZI2q1kpScRMoz92OcTHlrRwqLR9EvCY/t7ok8iYXhTYyfSnZr6Xwv96tQTCy4a1FSfAJJ0cChOBQeE0MtzimlEQPSsI1dY8xbRDNKFgg01D8OctL4ryWd738v7Dea5wM40jgw7RETpBPrpABXSPiqiEKHpGr+gdjZwX5835cD4no0vOdOcA/Snn+we6t6X+</latexit><latexit sha1_base64="caFM+jlPy1xIx45vy/zHYtRI7u8=">AAACJXicbZDLSgMxFIYzXmu9jbp0EyyCG8uMCLpQKIrgsoK9QFtKJs20oZlkSM4Ipe3LuPFV3LiwiODKVzHTFqqtBwI/3zmHk/8PYsENeN6Xs7S8srq2ntnIbm5t7+y6e/tloxJNWYkqoXQ1IIYJLlkJOAhWjTUjUSBYJejepv3KE9OGK/kIvZg1ItKWPOSUgEVN9+quWY8IdHTUjzsKlBziazyYwZBLIob4FM8Qlxy4hYNs0815eW9ceFH4U5FD0yo23VG9pWgSMQlUEGNqvhdDo080cCrYMFtPDIsJ7ZI2q1kpScRMoz92OcTHlrRwqLR9EvCY/t7ok8iYXhTYyfSnZr6Xwv96tQTCy4a1FSfAJJ0cChOBQeE0MtzimlEQPSsI1dY8xbRDNKFgg01D8OctL4ryWd738v7Dea5wM40jgw7RETpBPrpABXSPiqiEKHpGr+gdjZwX5835cD4no0vOdOcA/Snn+we6t6X+</latexit>

For absorption or emission of a photon:

Schrödinger’s Equation:

Page 2: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption

Announcements• Drill Set due by 11 a.m. Wednesday.

Reading Quiz due before Thursday’s lecture.

• Virtual Problem sessions tomorrow. If unable to attend any sessions, email a copy of your work on the assigned problems to your regular PS instructor prior to the next lecture. Also, you are welcome to attend more than one PS!

• Hand-in set #7 due Monday - Check calendar pages for problems. More info on how to hand it in coming Thursday.

• Exam #2 grades posted. 24 hour window for regrade requests opens tomorrow at 10 p.m.

• Feeling overwhelmed or having personal or technological issues that are making participation in PHYS 212 difficult? Please reach out to anyone on the PHYS 212 team. We’re here to help.

A biochemist [who used this QM stuff] you should know:

Dr. Emmett Chappelle (1925-2019) Dr. Chappelle was best known for his work on photosynthesis and bioluminescence (ATP-for all you bio nerds!). He discovered that even single-celled organisms such as algae, are photosynthetic and developed fluorescent tags now used to visually see cancer cells.

Page 3: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption

1. The Infinite Square Well

U

0 L x

U(x) = 1<latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit><latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit><latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit><latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit>

U(x) = 0<latexit sha1_base64="xcq7exko7lwfiuu8yfcwDXQT+Mk=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htoV1KNs22odnsmmTFsvRPePGgiFf/jjf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYK7W96tMpukK4V67gGp4BLRM3JxXI0eiVv7r9mKYRk4YKonXHxYnxM6IMp4JNSt1Us4TQERmwjqWSREz72ezeCTqxSh+FsbIlDZqpvycyEmk9jgLbGREz1IveVPzP66QmvPQzLpPUMEnni8JUIBOj6fOozxWjRowtIVRxeyuiQ6IINTaikg3BXXx5mTTPai6uuXfnlfp1HkcRjuAYquDCBdThFhrgAQUBz/AKb86D8+K8Ox/z1oKTzxzCHzifP/HRjpU=</latexit><latexit sha1_base64="xcq7exko7lwfiuu8yfcwDXQT+Mk=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htoV1KNs22odnsmmTFsvRPePGgiFf/jjf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYK7W96tMpukK4V67gGp4BLRM3JxXI0eiVv7r9mKYRk4YKonXHxYnxM6IMp4JNSt1Us4TQERmwjqWSREz72ezeCTqxSh+FsbIlDZqpvycyEmk9jgLbGREz1IveVPzP66QmvPQzLpPUMEnni8JUIBOj6fOozxWjRowtIVRxeyuiQ6IINTaikg3BXXx5mTTPai6uuXfnlfp1HkcRjuAYquDCBdThFhrgAQUBz/AKb86D8+K8Ox/z1oKTzxzCHzifP/HRjpU=</latexit><latexit sha1_base64="xcq7exko7lwfiuu8yfcwDXQT+Mk=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htoV1KNs22odnsmmTFsvRPePGgiFf/jjf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYK7W96tMpukK4V67gGp4BLRM3JxXI0eiVv7r9mKYRk4YKonXHxYnxM6IMp4JNSt1Us4TQERmwjqWSREz72ezeCTqxSh+FsbIlDZqpvycyEmk9jgLbGREz1IveVPzP66QmvPQzLpPUMEnni8JUIBOj6fOozxWjRowtIVRxeyuiQ6IINTaikg3BXXx5mTTPai6uuXfnlfp1HkcRjuAYquDCBdThFhrgAQUBz/AKb86D8+K8Ox/z1oKTzxzCHzifP/HRjpU=</latexit><latexit sha1_base64="xcq7exko7lwfiuu8yfcwDXQT+Mk=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htoV1KNs22odnsmmTFsvRPePGgiFf/jjf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYK7W96tMpukK4V67gGp4BLRM3JxXI0eiVv7r9mKYRk4YKonXHxYnxM6IMp4JNSt1Us4TQERmwjqWSREz72ezeCTqxSh+FsbIlDZqpvycyEmk9jgLbGREz1IveVPzP66QmvPQzLpPUMEnni8JUIBOj6fOozxWjRowtIVRxeyuiQ6IINTaikg3BXXx5mTTPai6uuXfnlfp1HkcRjuAYquDCBdThFhrgAQUBz/AKb86D8+K8Ox/z1oKTzxzCHzifP/HRjpU=</latexit>

U(x) = 1<latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit><latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit><latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit><latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit>

� ~22m

d2 (x)

dx2+ U(x) (x) = E (x)

<latexit sha1_base64="4fKLGXOfg1kl7oCyDGWvQUIExe4=">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</latexit><latexit sha1_base64="4fKLGXOfg1kl7oCyDGWvQUIExe4=">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</latexit><latexit sha1_base64="4fKLGXOfg1kl7oCyDGWvQUIExe4=">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</latexit><latexit sha1_base64="4fKLGXOfg1kl7oCyDGWvQUIExe4=">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</latexit>

Page 4: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption

CT1: Given a width, L, for the infinite square well, which of the following could be an allowed wavenumber, kn? Hint: think about λ for standing waves.

1) 𝜋/3L

3) 𝜋/L

5) 2𝜋/L

2) 𝜋/2L

4) 4𝜋/3L

6) 3𝜋/L

Page 5: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption

CT2: For the n=3 level of the infinite square well, at which of the following positions are you most likely to find the particle? Hint: draw ψ

1) L/6

3) L/3

5) L

2) L/4

4) L/2

6) 2L

U

0 L x

Page 6: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption

2. Absorption and Emission

Page 7: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption

Example (4.11, 4.12): For our PheT simulation, E1 = 0.32, E2=1.28, & E3=2.87 eV. If an electron in the n=3 level transitions to n=1, what is the wavelength of the

emitted photon?

Page 8: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption

CT3: Shown at right is an energy level diagram for an infinite square well. For which of the transitions will the emitted light have the greatest frequency? 1 2 3 4 5

n=1 E=E1

E=4E12

E=9E13

E=16E14

Energy

Page 9: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption

CT4: An electron is in the n=1 level. The electron encounters a photon with energy Eph=4 E1. What happens to the electron?

1) it absorbs the photon and jumps to the n=2 level. 2) it absorbs the photon and jumps to the n=3 level. 3) it has an energy of E=5 E1 and then jumps to n=3 when it encounters another photon with Eph=4 E1. 4) it jumps out of the well. 5) none of the above.

n=1 E=E1

E=4E12

E=9E13

E=16E14

Energy

Page 10: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption
Page 11: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption

3. Finite wells

U

0 L x

U(x) = 0<latexit sha1_base64="xcq7exko7lwfiuu8yfcwDXQT+Mk=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htoV1KNs22odnsmmTFsvRPePGgiFf/jjf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYK7W96tMpukK4V67gGp4BLRM3JxXI0eiVv7r9mKYRk4YKonXHxYnxM6IMp4JNSt1Us4TQERmwjqWSREz72ezeCTqxSh+FsbIlDZqpvycyEmk9jgLbGREz1IveVPzP66QmvPQzLpPUMEnni8JUIBOj6fOozxWjRowtIVRxeyuiQ6IINTaikg3BXXx5mTTPai6uuXfnlfp1HkcRjuAYquDCBdThFhrgAQUBz/AKb86D8+K8Ox/z1oKTzxzCHzifP/HRjpU=</latexit><latexit sha1_base64="xcq7exko7lwfiuu8yfcwDXQT+Mk=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htoV1KNs22odnsmmTFsvRPePGgiFf/jjf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYK7W96tMpukK4V67gGp4BLRM3JxXI0eiVv7r9mKYRk4YKonXHxYnxM6IMp4JNSt1Us4TQERmwjqWSREz72ezeCTqxSh+FsbIlDZqpvycyEmk9jgLbGREz1IveVPzP66QmvPQzLpPUMEnni8JUIBOj6fOozxWjRowtIVRxeyuiQ6IINTaikg3BXXx5mTTPai6uuXfnlfp1HkcRjuAYquDCBdThFhrgAQUBz/AKb86D8+K8Ox/z1oKTzxzCHzifP/HRjpU=</latexit><latexit sha1_base64="xcq7exko7lwfiuu8yfcwDXQT+Mk=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htoV1KNs22odnsmmTFsvRPePGgiFf/jjf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYK7W96tMpukK4V67gGp4BLRM3JxXI0eiVv7r9mKYRk4YKonXHxYnxM6IMp4JNSt1Us4TQERmwjqWSREz72ezeCTqxSh+FsbIlDZqpvycyEmk9jgLbGREz1IveVPzP66QmvPQzLpPUMEnni8JUIBOj6fOozxWjRowtIVRxeyuiQ6IINTaikg3BXXx5mTTPai6uuXfnlfp1HkcRjuAYquDCBdThFhrgAQUBz/AKb86D8+K8Ox/z1oKTzxzCHzifP/HRjpU=</latexit><latexit sha1_base64="xcq7exko7lwfiuu8yfcwDXQT+Mk=">AAAB73icbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htoV1KNs22odnsmmTFsvRPePGgiFf/jjf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYK7W96tMpukK4V67gGp4BLRM3JxXI0eiVv7r9mKYRk4YKonXHxYnxM6IMp4JNSt1Us4TQERmwjqWSREz72ezeCTqxSh+FsbIlDZqpvycyEmk9jgLbGREz1IveVPzP66QmvPQzLpPUMEnni8JUIBOj6fOozxWjRowtIVRxeyuiQ6IINTaikg3BXXx5mTTPai6uuXfnlfp1HkcRjuAYquDCBdThFhrgAQUBz/AKb86D8+K8Ox/z1oKTzxzCHzifP/HRjpU=</latexit>

U(x) = 1<latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit><latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit><latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit><latexit sha1_base64="DdOyM4xtXzGj27c+gJo/rnJBayI=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJeyKoBch6MVjBDcJJEuYncwmQ2Zn15neYAj5Di8eFPHqx3jzb5w8DppY0FBUddPdFaZSGHTdb2dldW19YzO3ld/e2d3bLxwc1kySacZ9lshEN0JquBSK+yhQ8kaqOY1Dyeth/3bi1wdcG5GoBxymPIhpV4lIMIpWCvzS0xm5Ji2hIhy2C0W37E5Blok3J0WYo9oufLU6CctirpBJakzTc1MMRlSjYJKP863M8JSyPu3ypqWKxtwEo+nRY3JqlQ6JEm1LIZmqvydGNDZmGIe2M6bYM4veRPzPa2YYXQUjodIMuWKzRVEmCSZkkgDpCM0ZyqEllGlhbyWsRzVlaHPK2xC8xZeXSe287Lll7/6iWLmZx5GDYziBEnhwCRW4gyr4wOARnuEV3pyB8+K8Ox+z1hVnPnMEf+B8/gBDQ5Ed</latexit>

d2 (x)

dx2=

2m

~2 + (U(x)� E) (x)<latexit sha1_base64="tIqfcvtk9PKkoSIr4FKhdkbuf+0=">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</latexit><latexit sha1_base64="tIqfcvtk9PKkoSIr4FKhdkbuf+0=">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</latexit><latexit sha1_base64="tIqfcvtk9PKkoSIr4FKhdkbuf+0=">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</latexit><latexit sha1_base64="tIqfcvtk9PKkoSIr4FKhdkbuf+0=">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</latexit>

U(x) = U0<latexit sha1_base64="ojNf671YuYMptppYLMp6PiFtVPY=">AAAB8XicbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htsV1KNs22odnskmTFsvRfePGgiFf/jTf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYKz141adTdIW8Hu6VK7iGZ0DLxM1JBXI0euWvbj+macSkoYJo3XFxYvyMKMOpYJNSN9UsIXREBqxjqSQR0342u3iCTqzSR2GsbEmDZurviYxEWo+jwHZGxAz1ojcV//M6qQkv/YzLJDVM0vmiMBXIxGj6PupzxagRY0sIVdzeiuiQKEKNDalkQ3AXX14mzbOai2vu3Xmlfp3HUYQjOIYquHABdbiFBnhAQcIzvMKbo50X5935mLcWnHzmEP7A+fwBUOSPXQ==</latexit><latexit sha1_base64="ojNf671YuYMptppYLMp6PiFtVPY=">AAAB8XicbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htsV1KNs22odnskmTFsvRfePGgiFf/jTf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYKz141adTdIW8Hu6VK7iGZ0DLxM1JBXI0euWvbj+macSkoYJo3XFxYvyMKMOpYJNSN9UsIXREBqxjqSQR0342u3iCTqzSR2GsbEmDZurviYxEWo+jwHZGxAz1ojcV//M6qQkv/YzLJDVM0vmiMBXIxGj6PupzxagRY0sIVdzeiuiQKEKNDalkQ3AXX14mzbOai2vu3Xmlfp3HUYQjOIYquHABdbiFBnhAQcIzvMKbo50X5935mLcWnHzmEP7A+fwBUOSPXQ==</latexit><latexit sha1_base64="ojNf671YuYMptppYLMp6PiFtVPY=">AAAB8XicbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htsV1KNs22odnskmTFsvRfePGgiFf/jTf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYKz141adTdIW8Hu6VK7iGZ0DLxM1JBXI0euWvbj+macSkoYJo3XFxYvyMKMOpYJNSN9UsIXREBqxjqSQR0342u3iCTqzSR2GsbEmDZurviYxEWo+jwHZGxAz1ojcV//M6qQkv/YzLJDVM0vmiMBXIxGj6PupzxagRY0sIVdzeiuiQKEKNDalkQ3AXX14mzbOai2vu3Xmlfp3HUYQjOIYquHABdbiFBnhAQcIzvMKbo50X5935mLcWnHzmEP7A+fwBUOSPXQ==</latexit><latexit sha1_base64="ojNf671YuYMptppYLMp6PiFtVPY=">AAAB8XicbVBNSwMxEJ2tX7V+VT16CRahXkpWBL0IRS8eK7htsV1KNs22odnskmTFsvRfePGgiFf/jTf/jWm7B219MPB4b4aZeUEiuDYYfzuFldW19Y3iZmlre2d3r7x/0NRxqijzaCxi1Q6IZoJL5hluBGsnipEoEKwVjG6mfuuRKc1jeW/GCfMjMpA85JQYKz141adTdIW8Hu6VK7iGZ0DLxM1JBXI0euWvbj+macSkoYJo3XFxYvyMKMOpYJNSN9UsIXREBqxjqSQR0342u3iCTqzSR2GsbEmDZurviYxEWo+jwHZGxAz1ojcV//M6qQkv/YzLJDVM0vmiMBXIxGj6PupzxagRY0sIVdzeiuiQKEKNDalkQ3AXX14mzbOai2vu3Xmlfp3HUYQjOIYquHABdbiFBnhAQcIzvMKbo50X5935mLcWnHzmEP7A+fwBUOSPXQ==</latexit>

Page 12: Lecture 16: Quantized Energies - Bucknell Universityphys211/sp2020/cal/materials/lect16.pdfLecture 16: Quantized Energies 1.Wavefunctions and Energy Levels 2.Emission & Absorption

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