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# Notes5 - Lesson 5 I Time Independent Schrdinger Equation A...

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Lesson 5 I. Time Independent Schrödinger Equation A. The first problem that we will consider is the case where the Hamiltonian is not a function of time. In classical mechanics, the time derivative of a physical quantity, A, is given by { } H A, t A dt dA + = For the Hamiltonian, this reduces to { } t H H H, t H dt dH = + = Thus, the Hamiltonian is a constant of the motion if the Hamiltonian doesn’t depend on time. 0 dt dH = For systems where the Hamiltonian is the total energy of the system, we have that constant Energy V T H = = + = A similar relationship exists in Quantum Mechanics (see 6.2) where the physical quantities are replaced by their corresponding operators and the Poisson brackets are replaced by the commutator which we will discuss later. [ ] H ˆ , A ˆ 1 t A ˆ dt A d i + = Here the expectation value of the energy is a constant of the motion. B. When the Hamiltonian is not a function of time, it is possible to separate the time and spatial parts of the wave function. The Time

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Dependent Schrödinger Equation can then be attacked using a differential equation solution technique called “Separation of Variables.” The first step is to write the wave function as a product of a function which only depends on space and a second function that only depends on time. In the case of a 1-dimensional spatial problem, we have ( 29 ( 29 ( 29 t T x Φ t x, ψ = We now substitute the equation into our Time Dependent Schrödinger Equation. ( 29 ( 29 ( 29 t x, ψ x H ˆ t x, ψ t i = ( 29 ( 29 ( 29 ( 29 ( 29 x Φ x H ˆ t T t t T x Φ i = It is important to remember that the Hamiltonian includes partial derivates and other operators that operate on the spatial variable x. Thus, a function of t can be moved to the left of the operator but not a function of x. A similar argument holds for the left-hand side of the equation where only functions of the spatial variable can be pulled outside the partial derivative with respect to time. ( 29 ( 29 ( 29 ( 29 ( 29 x Φ x H ˆ x Φ 1 t t T t T i = The left hand side of the equation contains only the variable t while the right hand side of the equation contains only the variable x. This equation must be true for any value of t or x. Since we could arbitrarily choose to just vary t or to just vary x, the only way that this equation will be valid is if the equation is equal to a constant which we will call E. ( 29 ( 29 ( 29 ( 29 ( 29 E x Φ x H ˆ x Φ 1 t t T t T i = =
We now have broken the Time Dependent Schrödinger Equation containing both x and t into two equations each involving only one variable. With some simple algebra, we have ( 29 ( 29 t T E i t d t T d - = Time Equation ( 29 ( 29 ( 29 x

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Notes5 - Lesson 5 I Time Independent Schrdinger Equation A...

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