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buraqenigma
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How can i prove that u(x)=exp(-x^{2}/2) is the eigenfunction of \hat{A} = \frac{d^{2}}{dx^{2}}-x^2
Is u(x)=exp(-x^{2}/2) an Eigenfunction of \hat{A}?
- Context: Graduate
- Thread starter buraqenigma
- Start date
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- Eigenfunctions Operator
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SUMMARY
The function u(x) = exp(-x²/2) is proven to be an eigenfunction of the operator \hat{A} = d²/dx² - x². This is established by verifying the equation \hat{A} u(x) = λ u(x), where λ is a constant. The discussion emphasizes the importance of applying the operator to the function and confirming the resulting expression aligns with the eigenvalue equation.
PREREQUISITES- Understanding of differential operators, specifically second derivatives.
- Familiarity with eigenfunctions and eigenvalues in quantum mechanics.
- Knowledge of exponential functions and their properties.
- Basic calculus skills for manipulating functions and derivatives.
- Study the properties of differential operators in quantum mechanics.
- Learn about the role of eigenfunctions in solving the Schrödinger equation.
- Explore the mathematical derivation of eigenvalues and eigenfunctions.
- Investigate the implications of the harmonic oscillator model in quantum physics.
Students and professionals in physics and mathematics, particularly those focusing on quantum mechanics and differential equations, will benefit from this discussion.
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