A particle is described by a wave function ψ(x)=Ae−αx2\psi(x)=$$Ae^{-\alpha x^2}$$, where AA and αα are real, positive constants. If the value of αα is increased, what effect does this have on (a) the particle’s uncertainty in position and (b) the particle’s uncertainty in momentum? Explain your answers.

Ch 40: Quantum Mechanics I: Wave Functions

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 40: Quantum Mechanics I: Wave Functions

Problem 4

Chapter 40, Problem 4

A particle is described by a wave function $ψ (x) = A e^{- α x^{2}}$ , where $A$ and $α$ are real, positive constants. If the value of $α$ is increased, what effect does this have on (a) the particle’s uncertainty in position and (b) the particle’s uncertainty in momentum? Explain your answers.

Verified step by step guidance

Step 1: Begin by understanding the wave function ψ(x) = A e^(-αx^2). This is a Gaussian wave function, which is commonly used in quantum mechanics to describe the probability amplitude of a particle's position. The parameter α determines the width of the Gaussian curve, and hence it is related to the spread of the particle's position.

Step 2: Recall the uncertainty principle, ΔxΔp ≥ ℏ/2, which states that the product of the uncertainties in position (Δx) and momentum (Δp) must be greater than or equal to ℏ/2. This principle will guide our analysis of how changes in α affect Δx and Δp.

Step 3: Analyze the effect of increasing α on the uncertainty in position (Δx). A larger α makes the Gaussian wave function narrower in position space, meaning the particle's position is more localized. Therefore, the uncertainty in position, Δx, decreases as α increases.

Step 4: Consider the relationship between position and momentum space. A narrower wave function in position space corresponds to a broader wave function in momentum space due to the Fourier transform relationship between the two. Thus, as α increases and Δx decreases, the uncertainty in momentum, Δp, increases.

Step 5: Conclude that increasing α decreases the uncertainty in position (Δx) while increasing the uncertainty in momentum (Δp). This behavior is consistent with the uncertainty principle, ΔxΔp ≥ ℏ/2, which remains satisfied throughout.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Wave Function and Probability Density

In quantum mechanics, a wave function, denoted as ψ(x), describes the quantum state of a particle. The square of the absolute value of the wave function, |ψ(x)|², gives the probability density of finding the particle at position x. This concept is fundamental for understanding how the behavior of particles is probabilistic rather than deterministic.

Uncertainty Principle

The Heisenberg Uncertainty Principle states that there is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously. Specifically, increasing the certainty in position (decreasing uncertainty in position) leads to an increase in uncertainty in momentum, and vice versa. This principle is crucial for analyzing the effects of changes in the wave function.

Gaussian Wave Packets

The given wave function ψ(x) = Ae^(-αx^2) represents a Gaussian wave packet, which is characterized by its bell-shaped curve. The parameter α controls the width of the wave packet; increasing α results in a narrower wave function, which implies a more localized particle position. This localization leads to a corresponding increase in the uncertainty of momentum, illustrating the trade-off described by the Uncertainty Principle.

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Related Practice

Textbook Question

An electron is moving as a free particle in the $- x$ -direction with momentum that has magnitude $4.50 \times 10^{- 24}$ kg*m/s. Let $k_{2} = 3 k_{1} = 3 k$ . At $t equals 0$ , the probability distribution function $∣ Ψ (x, t) ∣^{2}$ has a maximum at $x equals 0$ .

(a) What is the smallest positive value of $x$ for which the probability distribution function has a maximum at time $t = \frac{2 π}{ω}$ , where $ω = h k^{2} / 2 m$ ?

(b) From your result in part (a), what is the average speed with which the probability distribution is moving in the $+ x$ -direction?

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Textbook Question

Let $psi sub 1$ and $psi sub 2$ be two solutions of Eq. ( $40.23$ ) [ $- \frac{h^{2}}{2 m} \frac{d^{2} ψ (x)}{d x^{2}} + U (x) ψ (x) = E ψ (x)$ ] with energies $E sub 1$ and $E sub 2$ respectively, where $E sub 1 is not equal to E sub 2$ . Is $ψ = A ψ_{1} + B ψ_{2}$ , where $A$ and $B$ are nonzero constants, a solution to Eq. ( $40.23$ )? Explain your answer.

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Textbook Question

An electron is moving as a free particle in the $- x$ -direction with momentum that has magnitude $4.50 \times 10^{- 24}$ kg-m/s. What is the one-dimensional time-dependent wave function of the electron?

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Textbook Question

(a) Find the lowest energy level for a particle in a box if the particle is a billiard ball ( $m equals 0.20$ kg) and the box has a width of $1.3$ m, the size of a billiard table. (Assume that the billiard ball slides without friction rather than rolls; that is, ignore the rotational kinetic energy.)

(b) Since the energy in part (a) is all kinetic, to what speed does this correspond? How much time would it take at this speed for the ball to move from one side of the table to the other?

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Textbook Question

Consider a wave function given by $ψ (x) = A s i n k x$ , where $k = 2 π / λ$ and $A$ is a real constant.

(a) For what values of $x$ is there the highest probability of finding the particle described by this wave function? Explain.

(b) For which values of $x$ is the probability zero? Explain.

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Textbook Question

A free particle moving in one dimension has wave function $ψ (x, t) = A [e^{i (k x - ω t)} - e^{i (2 k x - 4 ω t)}]$ where $k$ and $v$ are positive real constants.

(a) At $t equals 0$ , what are the two smallest positive values of $x$ for which the probability function $∣ ψ (x, t) ∣^{2}$ is a maximum?

(b) Repeat part (a) for time $t = \frac{2 π}{ω}$ .

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