Ch 40: Quantum Mechanics I: Wave Functions

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 40: Quantum Mechanics I: Wave Functions

Problem 27

Chapter 40, Problem 27

(a) An electron with initial kinetic energy $32$ eV encounters a square barrier with height $41$ eV and width $0.25$ nm. What is the probability that the electron will tunnel through the barrier?
(b) A proton with the same kinetic energy encounters the same barrier. What is the probability that the proton will tunnel through the barrier?

Verified step by step guidance

Step 1: Begin by understanding the concept of quantum tunneling. Quantum tunneling occurs when a particle passes through a potential barrier that it classically does not have enough energy to overcome. The probability of tunneling is determined using the Schrödinger equation and depends on the particle's mass, energy, and the barrier's properties.

Step 2: For part (a), calculate the wave number inside the barrier for the electron using the formula:

k = \sqrt{\frac{2 m (V - E)}{ħ}}

, where

m

is the mass of the electron,

V

is the barrier height,

E

is the particle's energy, and

ħ

is the reduced Planck constant.

Step 3: Use the tunneling probability formula:

P = e^{-} (2 k a)

, where

a

is the width of the barrier and

k

is the wave number calculated in Step 2. Substitute the values for the electron's mass, energy, barrier height, and width to find the tunneling probability.

Step 4: For part (b), repeat the process for the proton. Since the proton has a much larger mass than the electron, its wave number

k

will be different. Use the same formula for

k

and substitute the proton's mass instead of the electron's mass.

Step 5: Calculate the tunneling probability for the proton using the same formula:

P = e^{-} (2 k a)

. Compare the probabilities for the electron and proton to understand the effect of mass on tunneling probability.

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Key Concepts

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

Quantum Tunneling

Quantum tunneling is a phenomenon in quantum mechanics where a particle has a probability of passing through a potential energy barrier, even if its energy is less than the height of the barrier. This occurs because particles exhibit wave-like properties, allowing them to exist in a superposition of states, which can lead to a non-zero probability of being found on the other side of the barrier.

Barrier Penetration Probability

The probability of a particle tunneling through a barrier can be calculated using the transmission coefficient, which depends on the height and width of the barrier as well as the energy of the particle. For a rectangular barrier, this probability decreases exponentially with increasing barrier height and width, making it less likely for particles with lower energy to tunnel through.

Mass and Energy Relationship

In quantum mechanics, the mass of a particle affects its wave function and, consequently, its tunneling probability. Heavier particles, like protons, have a shorter wavelength compared to lighter particles, like electrons, which influences their ability to tunnel through barriers. This means that for the same kinetic energy, a proton will generally have a lower tunneling probability than an electron when encountering the same barrier.

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Related Practice

Textbook Question

An electron with initial kinetic energy $6.0$ eV encounters a barrier with height $11.0$ eV. What is the probability of tunneling if the width of the barrier is (a) $0.80$ nm and (b) $0.40$ nm?

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Textbook Question

Recall that $(∣ ψ ∣^{2}) d x$ is the probability of finding the particle that has normalized wave function $ψ (x)$ in the interval $x$ to $x + d x$ . Consider a particle in a box with rigid walls at $x equals 0$ and $x = L$ . Let the particle be in the ground level and use $ψ_{n}$ as given in Eq. ( $40.35$ ) $ψ_{n} (x) = \sqrt{\frac{2}{L}} s i n [(\frac{n π x}{L})]$ where $n equals 1 comma 2 comma 3 comma dot dot dot$ .

(a) For which values of $x$ , if any, in the range from $0$ to $L$ is the probability of finding the particle zero?

(b) For which values of $x$ is the probability highest?

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Textbook Question

(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box of width $0.360$ nm.

(b) The electron makes a transition from the $n equals 1$ to $n equals 4$ level by absorbing a photon. Calculate the wavelength of this photon.

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Textbook Question

An electron is in a box of width $3.0 \times 10^{- 10}$ m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the $n equals 1$ level; (b) the $n equals 2$ level; (c) the $n equals 3$ level? In each case how does the wavelength compare to the width of the box?

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Textbook Question

An electron with initial kinetic energy $5.0$ eV encounters a barrier with height $U sub 0$ and width $0.60$ nm. What is the transmission coefficient if (a) $U sub 0 equals 7.0$ eV; (b) $U sub 0 equals 9.0$ eV; (c) $U sub 0 equals 13.0$ eV?

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Textbook Question

While undergoing a transition from the $n equals 1$ to the $n equals 2$ energy level, a harmonic oscillator absorbs a photon of wavelength $6.50$ $μ$ m. What is the wavelength of the absorbed photon when this oscillator undergoes a transition (a) from the $n equals 2$ to the $n equals 3$ energy level and (b) from the $n equals 1$ to the $n equals 3$ energy level?

(c) What is the value of $\sqrt{(k^{'} / m)}$ , the angular oscillation frequency of the corresponding Newtonian oscillator?

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