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1. Given two normalized and mutually orthogonal wave functions ψa(x)\psi_{a}(x) an...
May 7, 2024
Answer
The expectation value of the separation squared between two particles for each wave function is calculated using the integral of the probability density function times the separation squared.
Solution
a
For wave function ψd(x1,x2)=ψa(x1)ψb(x2) \psi_{\mathbf{d}}(x_{1}, x_{2}) = \psi_{a}(x_{1}) \psi_{b}(x_{2}) , the expectation value is calculated as: ⟨(x1−x2)2⟩=∫∫ψd∗(x1,x2)(x1−x2)2ψd(x1,x2)dx1dx2 \langle (x_{1} - x_{2})^{2} \rangle = \int \int \psi_{\mathbf{d}}^*(x_{1}, x_{2}) (x_{1} - x_{2})^{2} \psi_{\mathbf{d}}(x_{1}, x_{2}) dx_{1} dx_{2}
b
For wave function ψ+(x1,x2) \psi_{+}(x_{1}, x_{2}) , the expectation value is calculated similarly, but considering the cross terms that arise from the linear combination of the wave functions
c
For wave function ψ−(x1,x2) \psi_{-}(x_{1}, x_{2}) , the expectation value also includes cross terms, but with a minus sign, which may lead to cancellations in the integral
Key Concept
The expectation value of an observable in quantum mechanics is calculated by integrating the product of the complex conjugate of the wave function, the observable, and the wave function over all space.
Explanation
The expectation value gives the average outcome of a measurement of the observable for a particle described by a given wave function. In this case, the observable is the separation squared between two particles.
Answer
The completely antisymmetric wave function for a three-identical-particle system with states ψa(x)\psi_{a}(x), ψb(x)\psi_{b}(x), and ψc(x)\psi_{c}(x) is given by the Slater determinant formed by these states.
Solution
a
Constructing the Slater determinant: The antisymmetric wave function for a system of three fermions (particles that obey the Pauli exclusion principle) can be constructed using the Slater determinant. For the wave functions ψa(x)\psi_{a}(x), ψb(x)\psi_{b}(x), and ψc(x)\psi_{c}(x), the determinant is given by: Ψ(x1,x2,x3)=16∣ψa(x1)amp;ψb(x1)amp;ψc(x1)ψa(x2)amp;ψb(x2)amp;ψc(x2)ψa(x3)amp;ψb(x3)amp;ψc(x3)∣ \Psi(x_1, x_2, x_3) = \frac{1}{\sqrt{6}} \begin{vmatrix} \psi_{a}(x_1) & \psi_{b}(x_1) & \psi_{c}(x_1) \\ \psi_{a}(x_2) & \psi_{b}(x_2) & \psi_{c}(x_2) \\ \psi_{a}(x_3) & \psi_{b}(x_3) & \psi_{c}(x_3) \end{vmatrix}
b
Expanding the determinant: The wave function Ψ(x1,x2,x3)\Psi(x_1, x_2, x_3) is expanded as follows: Ψ(x1,x2,x3)=16(ψa(x1)ψb(x2)ψc(x3)−ψa(x1)ψc(x2)ψb(x3)+ψc(x1)ψa(x2)ψb(x3)−ψb(x1)ψa(x2)ψc(x3)+ψb(x1)ψc(x2)ψa(x3)−ψc(x1)ψb(x2)ψa(x3)) \Psi(x_1, x_2, x_3) = \frac{1}{\sqrt{6}} \left( \psi_{a}(x_1)\psi_{b}(x_2)\psi_{c}(x_3) - \psi_{a}(x_1)\psi_{c}(x_2)\psi_{b}(x_3) + \psi_{c}(x_1)\psi_{a}(x_2)\psi_{b}(x_3) - \psi_{b}(x_1)\psi_{a}(x_2)\psi_{c}(x_3) + \psi_{b}(x_1)\psi_{c}(x_2)\psi_{a}(x_3) - \psi_{c}(x_1)\psi_{b}(x_2)\psi_{a}(x_3) \right)
Key Concept
The Slater determinant is a mathematical tool used to construct antisymmetric wave functions for systems of identical fermions.
Explanation
The antisymmetric property ensures that the wave function changes sign upon the exchange of any two particles, which is a requirement for fermions due to the Pauli exclusion principle. The normalization factor 16\frac{1}{\sqrt{6}} ensures that the wave function is normalized if the individual states are normalized and orthogonal.
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