Question

Number of bound states for the system \( V(x) = -V_0 \delta(x-a), V_0>0 \) is:

• A. \( 1 \)
• B. \( 0 \)
• C. \( \infty \)
• D. \( \text{depends on } V_0 a^2 \)

Answer 1

The Correct Option is A

Step 1: Understanding the Potential

The potential in the given system is a Dirac delta function located at \( x = a \). The delta function potential is given by:

\[ V(x) = -V_0 \delta(x-a) \] where \( V_0 \) is a positive constant and \( \delta(x-a) \) is the Dirac delta function centered at \( x = a \).

Step 2: Schrödinger Equation for the System

The Schrödinger equation for a particle under this potential is given by:

\[ -\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x) \]

Substituting the given potential into the Schrödinger equation, we get: \[ -\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} - V_0 \delta(x-a) \psi(x) = E \psi(x) \] For bound states, we expect that \( E < 0 \) (since bound states have negative energy). Let \( E = -\epsilon \), where \( \epsilon > 0 \). 

Step 3: Behavior of the Wavefunction The solution to this equation can be divided into two regions: - For \( x \neq a \), the potential is zero, and the solution to the Schrödinger equation is: \[ \psi(x) = A e^{-\kappa |x-a|} \] where \( \kappa = \frac{\sqrt{2m\epsilon}}{\hbar} \). - At \( x = a \), the wavefunction will have a discontinuity due to the delta function. This discontinuity is found by integrating the Schrödinger equation around \( x = a \). 

Step 4: Discontinuity in the Derivative of the Wavefunction Integrating the Schrödinger equation around \( x = a \), we find that the derivative of the wavefunction has a discontinuity: \[ \psi'(a^+) - \psi'(a^-) = -\frac{2mV_0}{\hbar^2} \psi(a) \] For a bound state to exist, the wavefunction must satisfy the condition that the discontinuity in the derivative is non-zero. This condition can only be satisfied if there is exactly one bound state, as the delta function potential creates a single bound state. 

Conclusion: Since the potential \( V(x) = -V_0 \delta(x-a) \) is a single delta function, it can support only one bound state. Therefore, the number of bound states is:

\( 1 \)