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General Physics

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Modifications of the Bohr Theory – Elliptical Modifications of the Bohr Theory – Elliptical OrbitsOrbits

Sommerfeld extended the results to include Sommerfeld extended the results to include elliptical orbitselliptical orbits Retained the Retained the principle quantum numberprinciple quantum number, n, n Added the Added the orbital quantum numberorbital quantum number, , ℓℓ

ℓ ℓ ranges from 0 to n-1 in integer stepsranges from 0 to n-1 in integer steps All states with the same principle quantum All states with the same principle quantum

number are said to form a number are said to form a shellshell The states with given values of n and ℓ are said The states with given values of n and ℓ are said

to form a to form a subshellsubshell

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Modifications of the Bohr Theory – Zeeman Modifications of the Bohr Theory – Zeeman Effect and fine structureEffect and fine structure

Another modification was needed to account for the Another modification was needed to account for the Zeeman effectZeeman effect The Zeeman effect is the splitting of spectral lines in a strong The Zeeman effect is the splitting of spectral lines in a strong

magnetic fieldmagnetic field This indicates that the energy of an electron is slightly modified This indicates that the energy of an electron is slightly modified

when the atom is immersed in a magnetic fieldwhen the atom is immersed in a magnetic field A new quantum number, A new quantum number, mm ℓℓ,, called the called the orbital magnetic quantum orbital magnetic quantum

numbernumber, had to be introduced, had to be introduced

m m ℓℓ can vary from - ℓ to + ℓ in integer steps can vary from - ℓ to + ℓ in integer steps

High resolution spectrometers show that spectral lines are, in fact, High resolution spectrometers show that spectral lines are, in fact, two very closely spaced lines, even in the absence of a magnetic fieldtwo very closely spaced lines, even in the absence of a magnetic field This splitting is called This splitting is called fine structurefine structure Another quantum number, Another quantum number, mmss, called the , called the spin magnetic quantum spin magnetic quantum

number,number, was introduced to explain the fine structure was introduced to explain the fine structure

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28.5 de Broglie Waves28.5 de Broglie Waves

One of Bohr’s postulates was the angular momentum of One of Bohr’s postulates was the angular momentum of the electron is quantized, but there was no explanation the electron is quantized, but there was no explanation why the restriction occurredwhy the restriction occurred

de Broglie assumed that the electron orbit would be de Broglie assumed that the electron orbit would be stable only if it contained an integral number of electron stable only if it contained an integral number of electron wavelengthswavelengths

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28.6 Quantum Mechanics and the Hydrogen 28.6 Quantum Mechanics and the Hydrogen AtomAtom

One of the first great achievements of quantum mechanics was the One of the first great achievements of quantum mechanics was the solution of the wave equation for the hydrogen atomsolution of the wave equation for the hydrogen atom

The significance of quantum mechanics is that the quantum The significance of quantum mechanics is that the quantum numbers and the restrictions placed on their values arise directly numbers and the restrictions placed on their values arise directly from the mathematics and not from any assumptions made to make from the mathematics and not from any assumptions made to make the theory agree with experimentsthe theory agree with experiments

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Quantum Number SummaryQuantum Number Summary

The values of n can increase from 1 in The values of n can increase from 1 in integerinteger steps steps

The values of The values of ℓ can range from 0 to n-1 in integer stepsℓ can range from 0 to n-1 in integer steps

The values of The values of mm ℓℓ can range from -ℓ to ℓ in integer steps can range from -ℓ to ℓ in integer steps

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Spin Magnetic Quantum NumberSpin Magnetic Quantum Number

It is convenient to think of the It is convenient to think of the electron as spinning on its axiselectron as spinning on its axis

The electron is The electron is notnot physically spinningphysically spinning

There are There are two directionstwo directions for for the spinthe spin

Spin up, mSpin up, mss = = ½½ Spin down, mSpin down, mss = - = -½½

There is a slight energy There is a slight energy difference between the two difference between the two spins and this accounts for the spins and this accounts for the Zeeman effectZeeman effect

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