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The approximate ground state energy of the system from the linear variational principle. A variational calculation is done with the normalized trial wavefunction (x) = (a2-x²) for the one 4a 512 dimensional potential well, V(x)=√or a JAVO ∞ if x>a The ground state energisestimated foerendeavour.com Sh2 3ma 3h2 2ma2 (a) (b) (c) 3h2 5ma2 (d) 5h2 4ma2?
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The approximate ground state energy of the system from the linear vari...
Understanding the Variational Principle
The linear variational principle states that the expectation value of the Hamiltonian calculated with any normalized trial wavefunction provides an upper bound to the true ground state energy of the system.
Trial Wavefunction
For this problem, the trial wavefunction is given as:
- Psi(x) = (a² - x²) for -a < x="" />< />
- Psi(x) = 0 for |x| > a
This wavefunction is normalized over the interval [-a, a].
Expectation Value Calculation
1. Hamiltonian: The system is under a potential well where V(x) = 0 for |x| ≤ a and V(x) = ∞ for |x| > a.
2. Kinetic Energy: The kinetic energy operator is a second derivative with respect to x. Therefore, calculate the expectation value of the Hamiltonian using the trial wavefunction.
3. Integrate: The integration limits will be from -a to a. The calculations will provide the average energy.
Ground State Energy Estimation
After performing the calculation, the expectation value will yield an energy expression in terms of fundamental constants (h, m, and a):
- The ground state energy is approximated as: E ≈ 5h² / (4ma²)
Conclusion
The expected ground state energy for the given potential well using the variational method leads us to option (d) 5h² / (4ma²), which reflects the correct application of the variational principle and the chosen trial wavefunction. This method effectively provides an upper bound for the true ground state energy of the quantum system.
The linear variational principle states that the expectation value of the Hamiltonian calculated with any normalized trial wavefunction provides an upper bound to the true ground state energy of the system.
Trial Wavefunction
For this problem, the trial wavefunction is given as:
- Psi(x) = (a² - x²) for -a < x="" />< />
- Psi(x) = 0 for |x| > a
This wavefunction is normalized over the interval [-a, a].
Expectation Value Calculation
1. Hamiltonian: The system is under a potential well where V(x) = 0 for |x| ≤ a and V(x) = ∞ for |x| > a.
2. Kinetic Energy: The kinetic energy operator is a second derivative with respect to x. Therefore, calculate the expectation value of the Hamiltonian using the trial wavefunction.
3. Integrate: The integration limits will be from -a to a. The calculations will provide the average energy.
Ground State Energy Estimation
After performing the calculation, the expectation value will yield an energy expression in terms of fundamental constants (h, m, and a):
- The ground state energy is approximated as: E ≈ 5h² / (4ma²)
Conclusion
The expected ground state energy for the given potential well using the variational method leads us to option (d) 5h² / (4ma²), which reflects the correct application of the variational principle and the chosen trial wavefunction. This method effectively provides an upper bound for the true ground state energy of the quantum system.
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The approximate ground state energy of the system from the linear variational principle. A variational calculation is done with the normalized trial wavefunction (x) = (a2-x²) for the one 4a 512 dimensional potential well, V(x)=√or a JAVO ∞ if x>a The ground state energisestimated foerendeavour.com Sh2 3ma 3h2 2ma2 (a) (b) (c) 3h2 5ma2 (d) 5h2 4ma2?
Question Description
The approximate ground state energy of the system from the linear variational principle. A variational calculation is done with the normalized trial wavefunction (x) = (a2-x²) for the one 4a 512 dimensional potential well, V(x)=√or a JAVO ∞ if x>a The ground state energisestimated foerendeavour.com Sh2 3ma 3h2 2ma2 (a) (b) (c) 3h2 5ma2 (d) 5h2 4ma2? for Chemistry 2024 is part of Chemistry preparation. The Question and answers have been prepared according to the Chemistry exam syllabus. Information about The approximate ground state energy of the system from the linear variational principle. A variational calculation is done with the normalized trial wavefunction (x) = (a2-x²) for the one 4a 512 dimensional potential well, V(x)=√or a JAVO ∞ if x>a The ground state energisestimated foerendeavour.com Sh2 3ma 3h2 2ma2 (a) (b) (c) 3h2 5ma2 (d) 5h2 4ma2? covers all topics & solutions for Chemistry 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The approximate ground state energy of the system from the linear variational principle. A variational calculation is done with the normalized trial wavefunction (x) = (a2-x²) for the one 4a 512 dimensional potential well, V(x)=√or a JAVO ∞ if x>a The ground state energisestimated foerendeavour.com Sh2 3ma 3h2 2ma2 (a) (b) (c) 3h2 5ma2 (d) 5h2 4ma2?.
The approximate ground state energy of the system from the linear variational principle. A variational calculation is done with the normalized trial wavefunction (x) = (a2-x²) for the one 4a 512 dimensional potential well, V(x)=√or a JAVO ∞ if x>a The ground state energisestimated foerendeavour.com Sh2 3ma 3h2 2ma2 (a) (b) (c) 3h2 5ma2 (d) 5h2 4ma2? for Chemistry 2024 is part of Chemistry preparation. The Question and answers have been prepared according to the Chemistry exam syllabus. Information about The approximate ground state energy of the system from the linear variational principle. A variational calculation is done with the normalized trial wavefunction (x) = (a2-x²) for the one 4a 512 dimensional potential well, V(x)=√or a JAVO ∞ if x>a The ground state energisestimated foerendeavour.com Sh2 3ma 3h2 2ma2 (a) (b) (c) 3h2 5ma2 (d) 5h2 4ma2? covers all topics & solutions for Chemistry 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The approximate ground state energy of the system from the linear variational principle. A variational calculation is done with the normalized trial wavefunction (x) = (a2-x²) for the one 4a 512 dimensional potential well, V(x)=√or a JAVO ∞ if x>a The ground state energisestimated foerendeavour.com Sh2 3ma 3h2 2ma2 (a) (b) (c) 3h2 5ma2 (d) 5h2 4ma2?.
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