Variation method for the harmonic oscillatorUsually we use a trial function that contains one or more adjustable parameters, and minimizethe energy with respect to the parameters. We can use the variation method in this way to findthe harmonic osillator ground state, using the trial function i = e-ax?k-amHy =一y+一2mV元/0h'αV元/a+[k-a'ty"Hyjidx2α2m2mAlso*dx=/元/a-[hαK40(i)4m
Variation method for the harmonic oscillator Usually we use a trial function that contains one or more adjustable parameters, and minimize the energy with respect to the parameters. We can use the variation method in this way to find the harmonic oscillator ground state, using the trial function 𝜓෨ = e − 1 2 𝛼𝑥 2 . 2 2 2 1 2 2 2 H x k m m 2 2 2 * 1 / d / 2 2 2 H x k m m 2 2 2 2 1 2 4 4 4 H k E k m m m ෨�� Also ∗ 𝜓෨ d𝑥 = 𝜋/𝛼
E_(H)Kh'αh'ααh(il)4m4α2mmTo find the lowest energy, we minimize with respect to a andα too bigeasily find that aα = Vkm/h, as before.The first term in E is the expectation value <T) of the kineticαtposmallenergy,while the second term is theexpectationvalue<V)ofthe potential energy. If α is large, we get a sharply peaked which has a low potential energy but a high kinetic energy. Ifα is small,thewavefunction is broad and varies slowlywithx, so <T) is small, but it extends into regions where thepotential energy is high
2 2 2 2 1 2 4 4 4 H k E k m m m To find the lowest energy, we minimize with respect to 𝛼 and easily find that 𝛼 = 𝑘𝑚/ℏ, as before. The first term in 𝐸෨ is the expectation value 𝑇 of the kinetic energy, while the second term is the expectation value 𝑉 of the potential energy. If 𝛼 is large, we get a sharply peaked 𝜓෨ which has a low potential energy but a high kinetic energy. If 𝛼 is small, the wave function is broad and varies slowly with x, so 𝑇 is small, but it extends into regions where the potential energy is high
LinearcombinationofatomicorbitalsUsually we choose a trial wavefunction that has one or more adjustable parameters in it, andchoose values for the parameters that minimize the energy. An very important type of trialfunctionisthelinear combinationofatomic orbitals(L.C.A.O.),whichwecanillustrateforthehydrogenmoleculeionThe wavefunction for an individual hydrogen atom is the ls orbital; for ahydrogenmoleculeionwetrythewavefunctionji=c.Sa+C,Sbwhere s.is the normalized ls orbital for atom a and s, for atom b, and c.and c, arenumerical coefficientsthatweshalladjusttominimizetheenergy.Thisfunctionbehaveslike s,near nucleus a, where s, is small, and like s, near nucleus b
Linear combination of atomic orbitals Usually we choose a trial wavefunction that has one or more adjustable parameters in it, and choose values for the parameters that minimize the energy. An very important type of trial function is the linear combination of atomic orbitals (L.C.A.O.), which we can illustrate for the hydrogen molecule ion. The wavefunction for an individual hydrogen atom is the 1s orbital; for a hydrogen molecule ion we try the wavefunction where sa is the normalized 1s orbital for atom a and sb for atom b, and ca and cb are numerical coefficients that we shall adjust to minimize the energy. This function behaves like sa near nucleus a, where sb is small, and like sb near nucleus b. a a b b c s c s