Position Space And Momentum Space | Santa Fe College - Edubirdie

University:
Santa Fe College
Course:
PHY 2048 | General Physics 1 with Calculus
Academic year:

2024
Views:

303

Pages:

2
Author:

Jabin F.

Dirac Delta function | 1. Plane wave type Plane wave type | 2. Dirac Delta Gaussian function | 3. Gaussian function Constant | 4. Constant e^(-x^2 / (2σ^2)) | 4. e^(-p^2 / (2σ^2)) |ψ(x)|^2 → position probability density | 4. |Φ(p)|^2 → momentum probability density |ψ(x)|^2 dx → The probability of finding the position of the particle lies in the small space x to x+dx | 5. |Φ(p)|^2 dp → The probability of measuring the sharp value or particular momentum of the particle lies in the small interval p to p+dp ✰ = x [multiplicative nature] | ✰ = -iħ∇ K = E = p^2 / (2m) = -ħ^2 / (2m) * d^2 / dx^2 Momentum operators and any function of momentum operators have multiplicative nature.

= ∫_{-∞}^{∞} ψ*(x) (-iħ * d / dx) ψ(x) dx = -iħ ∫_{-∞}^{∞} ψ*(x) (dψ(x) / dx) dx = -iħ [ψ*(x) ψ(x)]_{-∞}^{∞} + iħ ∫_{-∞}^{∞} ψ(x) (dψ*(x) / dx) dx

= iħ ∫_{-∞}^{∞} ψ(x) (dψ*(x) / dx) dx If ψ(x) is normalized to unity,

= ∫_{-∞}^{∞} |ψ(p)|^2 dp It includes the following: Momentum Operator: The momentum operator is defined as -iħ * d / dx in position space. Multiplicative Nature: Any function of the momentum operator also has a multiplicative nature. Expectation Value of Momentum: The expectation value of momentum is calculated using the integral of the wave function multiplied by the momentum operator. Normalization: If the wave function is normalized, the expectation value of momentum can be simplified to the integral of the squared magnitude of the wave function in momentum space.

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