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4_Schrodinger

# 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
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 )
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
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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