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The eigenfunctions of Hermitian Operators are orthonormal (orthogonal and normalizable). a. Prove the eigenfunctions of the Hamiltonian Operator for a particle in a box that extends from x = 0 to x = a: плх w.(x) = are orthonormal by integrating a pair of functions, y.(x) and y.(x), with n = m in one case and n m in another. b. For the ground state of a particle in a box, use the momentum and position operators to show that the expectation values are 0 for momentum and - for position, by evaluating the resulting integrals. c. Calculate the uncertainty in x and p (0x and Op) for the lowest energy state of a particle in a box by taking the standard deviation of x and p as a measure of their uncertainty: Ar =0, = (x*)-(x)* and Ap=0, =Kp*)-(p)* Is the product you obtain for xOp consistent with the Heisenberg Uncertainty Principle?

The eigenfunctions of Hermitian Operators are orthonormal (orthogonal and normalizable). a. Prove the eigenfunctions of the Hamiltonian Operator for a particle in a box that extends from x = 0 to x = a: плх w.(x) = are orthonormal by integrating a pair of functions, y.(x) and y.(x), with n = m in one case and n m in another. b. For the ground state of a particle in a box, use the momentum and position operators to show that the expectation values are 0 for momentum and - for position, by evaluating the resulting integrals. c. Calculate the uncertainty in x and p (0x and Op) for the lowest energy state of a particle in a box by taking the standard deviation of x and p as a measure of their uncertainty: Ar =0, = (x*)-(x)* and Ap=0, =Kp*)-(p)* Is the product you obtain for xOp consistent with the Heisenberg Uncertainty Principle?

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The eigenfunctions of Hermitian Operators are orthonormal (orthogonal and normalizable). a. Prove the eigenfunctions of the Hamiltonian Operator for a particle in a box that extends from x = 0 to x = a: плх y.(x) = are orthonormal by integrating a pair of functions, y,(x) and y.(x), with n = m in one case and n m in another. b. For the ground state of a particle in a box, use the momentum and position operators to show that the expectation values are 0 for momentum and - for position, by evaluating the resulting integrals. c. Calculate the uncertainty in x and p (0x and Op) for the lowest energy state of a particle in a box by taking the standard deviation of x and p as a measure of their uncertainty: Ar =0, = (x*)-(x)* and Ap=o,=Kp*)-(p)* Is the product you obtain for OxOp consistent with the Heisenberg Uncertainty Principle?
Transcribed Image Text:The eigenfunctions of Hermitian Operators are orthonormal (orthogonal and normalizable). a. Prove the eigenfunctions of the Hamiltonian Operator for a particle in a box that extends from x = 0 to x = a: плх y.(x) = are orthonormal by integrating a pair of functions, y,(x) and y.(x), with n = m in one case and n m in another. b. For the ground state of a particle in a box, use the momentum and position operators to show that the expectation values are 0 for momentum and - for position, by evaluating the resulting integrals. c. Calculate the uncertainty in x and p (0x and Op) for the lowest energy state of a particle in a box by taking the standard deviation of x and p as a measure of their uncertainty: Ar =0, = (x*)-(x)* and Ap=o,=Kp*)-(p)* Is the product you obtain for OxOp consistent with the Heisenberg Uncertainty Principle?
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