Problem 59 - University Physics with Modern Physics

Question

A fellow student proposes that a possible wave function for a free particle with mass \(m\) (one for which the potential-energy function \(U(x)\) is zero) is $$\psi(x)=\left\\{\begin{array}{ll}{e^{+\kappa x},} & {x<0} \\ {e^{-\kappa x},} & {x \geq 0}\end{array}\right.$$ where \(\kappa\) is a positive constant. (a) Graph this proposed wave function. (b) Show that the proposed wave function satisfies the Schrodinger equation for \(x < 0\) if the energy is \(E=-\hbar^{2} \kappa^{2} / 2 m-\) that is, if the energy of the particle is negative. (c) Show that the proposed wave function also satisfies the Schrodinger equation for \(x \geq 0\) with the same energy as in part (b). (d) Explain why the proposed wave function is nonetheless not an acceptable solution of the Schrodinger equation for a free particle. (Hint: What is the behavior of the function at \(x=0 ?\) ) It is in fact impossible for a free particle (one for which \(U(x)=0 )\) to have an energy less than zero.

Step-by-Step Solution

Answer

The proposed wave function is not acceptable because it implies negative energy, which is impossible for a free particle.

Step 1: Plot the Wave Function

The proposed wave function is split into two regions, with different expressions for each: For \(x < 0\), the wave function \(\psi(x) = e^{\kappa x}\) increases exponentially towards positive infinity as \(x\) decreases. For \(x \geq 0\), the wave function \(\psi(x) = e^{-\kappa x}\) decreases exponentially towards zero as \(x\) increases. Plotting these, at \(x = 0\), the function is continuous and forms a smooth transition from increasing to decreasing.

Step 2: Verify the Schrödinger Equation for \(x < 0\)

We start by finding the second derivative of \(\psi(x) = e^{\kappa x}\): \[ \frac{d^2\psi}{dx^2} = \kappa^2 e^{\kappa x} \]Applying the time-independent Schrödinger equation, which for \(U(x) = 0\) is \[ -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E\psi \] Substitute the second derivative: \[ -\frac{\hbar^2}{2m}(\kappa^2 e^{\kappa x}) = Ee^{\kappa x} \] Simplifying, we find \( E = -\frac{\hbar^2 \kappa^2}{2m} \), which is negative.

Step 3: Verify the Schrödinger Equation for \(x \geq 0\)

Now consider \(\psi(x) = e^{-\kappa x}\) for \(x \geq 0\). The second derivative is: \[ \frac{d^2\psi}{dx^2} = \kappa^2 e^{-\kappa x} \] Substitute this into the Schrödinger equation: \[ -\frac{\hbar^2}{2m}(\kappa^2 e^{-\kappa x}) = Ee^{-\kappa x} \] This matches the earlier expression, yielding \(E = -\frac{\hbar^2 \kappa^2}{2m}\), which still results in a negative energy.

Step 4: Discuss the Acceptability of the Wave Function

Despite satisfying the Schrödinger equation with negative energy, this wave function cannot represent a free particle because free particles do not have potential energy, making negative total energy impossible. At \(x = 0\), the wave function's slope changes abruptly, indicating a discontinuity in the derivative which implies infinite momentum expectation value, violating physical laws.

Key Concepts

Wave function

In quantum mechanics, the wave function is a central concept that describes the quantum state of a particle. It provides information about the probability amplitude of a particle's position and state. For a free particle, a potential energy function, \(U(x)\), is zero everywhere, which implies that the particle is not influenced by external forces. The given wave function is piecewise, and though continuous at \(x = 0\), it consists of two different exponential expressions. This function suggests the behavior of a free particle on either side of the origin, making it a subject of interest in solving the Schrödinger equation for such cases.

Schrödinger equation

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the wave function evolves over time. For a free particle, the time-independent Schrödinger equation is written as: \[ -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E\psi \]This equation relates the second derivative of the wave function to the energy of the particle, showing how the potential-free environment allows for specific energy values. Solving this equation allows us to understand the nature of quantum states available to particles under different conditions, including those with negative energies in some unusual or theoretical constructs.

Free particle

A free particle, in the realm of quantum mechanics, is an idealized particle that moves without any external potential influencing its motion. The condition \(U(x) = 0\) describes such scenarios where the particle is entirely unconstrained. The wave function described models a free particle across two regions, \(x < 0\) and \(x \geq 0\), indicative of its supposed behavior without any restrictive energies externally applied. However, as the solution explored, a true free particle cannot possess a negative energy, questioning the validity of this wave function as a correct representation.

Negative energy

Negative energy in quantum mechanics often appears as a peculiar concept but underpins important theoretical discussions. For the proposed wave function, calculations reveal an energy result of \(E = -\frac{\hbar^2 \kappa^2}{2m}\) This negative energy inherently conflicts with the realistic scenario for a free particle, which should exhibit zero potential indeed. Though an interesting exploration, it underlines a scenario not typically possible in the physical realm because negative energy suggests a bound or potential condition. Thus, though mathematically solved, such results void of physical appeal for free particles, denote its impracticality.

Continuity in wave functions

Continuity of a wave function is crucial to ensuring viable solutions in quantum mechanics. At \(x = 0\), the wave function transitions smoothly from increasing to decreasing. While this appears continuous, scrutiny shows the derivative of this wave function is not continuous. Discontinuity in the derivative implies sudden changes in the momentum of the particle, suggesting infinite expectations that are non-physical. This problem in continuity therefore indicates incompleteness or unrealistic conditions for this wave function as a valid representation for a free particle, despite satisfying initial conditions of the Schrödinger equation in distinct cases.