Question

A system having Hamiltonian \( \hat{H} \) follows the eigenvalue equation, \( \hat{H} \psi_n = E_n \psi_n \), with \[ E_n = \left(n + \frac{1}{2}\right) \] If the state of the system is prepared as, \[ \Psi = N(\psi_1 + \psi_2 + \psi_3 - \psi_4 - \psi_5), \] where N is the normalization constant, then the expectation value of energy is

• A. -0.5
• B. -2.5
• C. 3.5
• D. 17.5

Hint:

In quantum mechanics, the expectation value of energy is calculated by summing the energy eigenvalues weighted by the coefficients of the state in that basis.

Answer 1

The Correct Option is C

Step 1: Understanding the expectation value of energy.
The expectation value of energy is given by: \[ \langle E \rangle = \langle \Psi | \hat{H} | \Psi \rangle \] Since the Hamiltonian \( \hat{H} \) is diagonal in the energy eigenstate basis, the energy expectation value can be computed as: \[ \langle E \rangle = N^2 \left( E_1 + E_2 + E_3 - E_4 - E_5 \right) \] where \( E_n \) is the energy associated with the state \( \psi_n \), given by \( E_n = \left(n + \frac{1}{2}\right) \).
Step 2: Calculating the individual energies.
Using the given formula for \( E_n \), we compute: \[ E_1 = \left(1 + \frac{1}{2}\right) = 1.5, \quad E_2 = \left(2 + \frac{1}{2}\right) = 2.5, \quad E_3 = \left(3 + \frac{1}{2}\right) = 3.5 \] \[ E_4 = \left(4 + \frac{1}{2}\right) = 4.5, \quad E_5 = \left(5 + \frac{1}{2}\right) = 5.5 \] Step 3: Substituting into the expectation value formula.
Now, substitute the energies into the expectation value equation: \[ \langle E \rangle = N^2 \left( 1.5 + 2.5 + 3.5 - 4.5 - 5.5 \right) \] \[ \langle E \rangle = N^2 \times (3.5) \] Since the normalization constant \( N^2 \) ensures the state is normalized, we have \( N^2 = 1 \). Step 4: Conclusion.
Thus, the expectation value of energy is \( 3.5 \). Final Answer: \[ \boxed{3.5} \]