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4_Schrodinger - IV Solving the Time Independent Schrdinger...

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IV. Solving the Time Independent Schrödinger Equation Griffiths Chapters 2 and 4 1
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A. The Operator Approach to the Harmonic Oscillator The harmonic oscillator is represented by the force equation F = -kx. This gives a potential energy of V(x) = -kx 2 /2. Many potentials will reduce to this form for small values of x. Therefore, as a first approximation, many physical phenomena can be understood using the harmonic potential. Griffiths 2.3 2
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V ( x ) = V ( x 0 ) + V ( x 0 )( x x 0 ) + 1 2 V ( x 0 )( x x 0 ) 2 + · · · Taylor expand V(x) about the point x 0 . drop constant term... doesn’t change the force x 0 is a minimum so the first derivative at x 0 is zero as long as (x-x 0 ) is small, higher order terms will be negligible For our quantum situation, we want to solve the Schrödinger equation for the potential k = V’’(x 0 ) ω = k m 3 A. The Operator Approach to the Harmonic Oscillator V ( X ) = 1 2 m ω 2 X 2
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The higher energy solutions are The time-independent Schrödinger equation will be This has an infinite number of discrete solutions. The ground state (lowest energy) of this quantum oscillator will be ψ 0 ( x ) = m ω π 1 / 4 e m ω 2 x 2 E 0 = 1 2 ω ground state wave function ground state energy ψ n ( x ) = A n ( a + ) n ψ 0 ( x ) E n = n + 1 2 ω 4 A. The Operator Approach to the Harmonic Oscillator 2 2 m d 2 ψ ( x ) dx 2 + 1 2 m ω 2 X 2 ψ ( x ) = E ψ ( x )
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The higher energy solutions are ψ n ( x ) = A n ( a + ) n ψ 0 ( x ) E n = n + 1 2 ω normalization constant So if we know what the raising operator a + is, then we can find all of the higher energy wave functions and their corresponding energies. a + and the related lowering operator a - are called ladder operators . See board IV.A.1 5 A. The Operator Approach to the Harmonic Oscillator
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The number operator is defined as N = a + a - . See board IV.A.2 N ψ n ( x ) = a + a ψ n ( x ) = n ψ n ( x ) The ladder operators can be used to normalize the wave functions of the harmonic oscillator. ψ n ( x ) = 1 n ! ( a + ) n ψ 0 ( x ) E n = n + 1 2 ω and normalized See board IV.A.3 6 A. The Operator Approach to the Harmonic Oscillator
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B. The Hydrogen Atom The hydrogen atom is one of the few, realistic, systems that can be solved analytically. The prescription is to write the time-independent Schrödinger equation in spherical coordinates and solve it using a Coulomb potential. The math goes on a long time but involves mostly just separation of variables. We use special functions such as Legendre polynomials, spherical harmonics and Laguerre polynomials to get our answer.
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