For a particle in a finite potential well of width L and depth U 0 , what is the ratio of the probability Prob(in δx at x=L+η) to the probability Prob(in δx at x = L)?

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem in a finite potential. Well, with a width of two nanometers and a depth of 20 electron volts determine the probability ratio between finding the particle within a small interval. Delta X at a position X equals L plus 0.5 theta and finding it within the same interval at the position X equals L here, Ada is a small distance outside the well awesome. So it appears for this particular problem we're ultimately trying to figure out what the probability ratio is for this particular problem. So now that we know that we're solving for the probability ratio, let's read off our multiple choice answers to see what our final answer might be. A is 0.135 B is 0.607 C is 0.368 and D is 0.821. OK. So first off, let us recall and note that in the classically forbidden region of A potential. Well, the wave function will decrease exponentially and can be written as the fly equation. And let's call this equation I an equation I states that and I'm gonna, you, I'm gonna say si, but not every single time that I'm saying I, this is a capital C I to be precise. So I of X is equal to si edge and I'm gonna use a capital E to denote edge. So si edge multiplied by E to the power of negative X minus L. And then we need to divide X minus L by Ada Awesome. And this is important to note that this is for X is greater than or equal to L Awesome. So now we need to recall and use the equation to help us solve for the probability of finding a particle in the small interval delta X in X which let's write this as the probability. And I'm gonna use capital P to denote probability of N delta X at X is equal to the absolute value of SI of X. And don't forget that this absolute value or so it's the absolute value of I of X and this value is squared delta X fantastic. So therefore, we could write our ratio equation as the probability in delta X at X equals L plus 0.58 divided by the probability uh the probability in delta X at X equals L. And this equation is all equal to the absolute value of SI of L plus 0.5 ada square delta X divided by the absolute value of SY of L. So this is SI of L squared delta X which this is all equal to the absolute value of SI of L plus 0.5 A fantastic. And don't forget that this value is squared divided by the absolute value of SI of L squared fantastic. And let us call this equation I I fantastic. So now moving right along here, we need to note and let's write this note in blue that at the edge of the forbidden region. So this is that X equals L. This will mean that s that SI of L is equal to si edge fantastic. So with that in mind, we can go ahead and write that at X equals L plus 0.5 Ada, we can go ahead and write that sigh of L plus 0.5 A is equal to sigh edge multiplied by E to the power of negative 0.5 fantastic. And once again, this is sy edge Awesome. So si, so this is equal to si edge multiplied by E to the power of negative 0.5. So now when we substitute these values back into our equation, I I, we can state that the probability of in delta X at X equals L plus 0.5 ada and this is divided by the probability in delta X at X is equal to bracket sigh edge multiplied by E to the power of negative 0.5 squared divided by sigh edge square is all equal to when you simplify E to the power of negative one, which when we round to three decimal places, we will find that E to the power of negative one is approximately equal to 0.368. And that is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be the letter C which once again is 0.368. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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Problem 36

Textbook Question

For a particle in a finite potential well of width L and depth U₀, what is the ratio of the probability Prob(in δx at x=L+η) to the probability Prob(in δx at x = L)?

Verified step by step guidance

Step 1: Recognize that the problem involves a quantum mechanical particle in a finite potential well. The probability density is related to the square of the wavefunction, |ψ(x)|², at the respective positions.

Step 2: For a finite potential well, the wavefunction inside the well (x ≤ L) is typically sinusoidal, while outside the well (x > L) it decays exponentially. The general form of the wavefunction outside the well is ψ(x) = A * e^(-κx), where κ = sqrt(2m(U₀ - E))/ħ, m is the particle's mass, U₀ is the well depth, E is the particle's energy, and ħ is the reduced Planck constant.

Step 3: To find the ratio of probabilities, calculate the square of the wavefunction at the two positions: Prob(in δx at x=L+η) ∝ |ψ(L+η)|² and Prob(in δx at x=L) ∝ |ψ(L)|². Using the exponential decay form of the wavefunction, |ψ(L+η)|² = |A|² * e^(-2κ(L+η)) and |ψ(L)|² = |A|² * e^(-2κL).

Step 4: Divide the probability at x=L+η by the probability at x=L to find the ratio: Prob(in δx at x=L+η) / Prob(in δx at x=L) = e^(-2κ(L+η)) / e^(-2κL). Simplify the expression to get e^(-2κη).

Step 5: Conclude that the ratio of probabilities depends only on the exponential decay factor e^(-2κη), where κ is determined by the particle's mass, the well depth U₀, and the energy E. This shows how the probability decreases as the particle moves further outside the well.

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Key Concepts

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

Finite Potential Well

A finite potential well is a region where a particle experiences a potential energy lower than its surroundings, allowing for bound states. The well has a specific width (L) and depth (U0), which influence the energy levels and wave functions of particles within it. Understanding the characteristics of the potential well is crucial for analyzing the behavior of quantum particles, particularly in terms of their probability distributions.

Quantum Probability Density

In quantum mechanics, the probability density describes the likelihood of finding a particle in a specific region of space. It is derived from the square of the wave function's amplitude, which represents the particle's state. The probability of locating the particle in a small interval (δx) can be calculated by integrating the probability density over that interval, making it essential for comparing probabilities at different positions within the potential well.

Boundary Conditions

Boundary conditions are constraints that the wave function must satisfy at the edges of the potential well. For a finite potential well, these conditions determine how the wave function behaves at the boundaries (x = 0 and x = L). They are critical for solving the Schrödinger equation and finding the allowed energy levels and corresponding wave functions, which directly affect the probability calculations at specific points within the well.

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