A particle of mass m in a one-dimensional box has the...

A particle of mass $m$ in a one-dimensional box has the following wave function in the region $x=0$ to $x=L:$ $$\Psi(x, t)=\frac{1}{\sqrt{2}} \psi_{1}(x) e^{-i E_{1} t / \hbar}+\frac{1}{\sqrt{2}} \psi_{3}(x) e^{-i E_{3} t / \hbar}$$ Here $\psi_{1}(x)$ and $\psi_{3}(x)$ are the normalized stationary-state wave functions for the $n=1$ and $n=3$ levels, and $E_{1}$ and $E_{3}$ are the energies of these levels. The wave function is zero for $x<0$ and for $x>L$ (a) Find the value of the probability distribution function at $x=L / 2$ as a function of time. (b) Find the angular frequency at which the probability distribution function oscillates.

Key Concepts

Particle in a One-Dimensional Box

This concept refers to a quantum mechanical model where a particle is confined to move within a region of space bounded by impenetrable barriers. The potential is zero inside the box and infinite outside, leading to discrete energy levels and well-defined stationary states. This model is fundamental for understanding quantization and the behavior of particles in confined spaces.

Stationary States and Energy Quantization

Stationary states are solutions of the time-independent Schrödinger equation for systems with time-independent Hamiltonians. In the context of the one-dimensional box, each stationary state corresponds to a specific energy level (quantized energy) and a distinct spatial wave function. The energies are determined by the quantum number, and the corresponding wave functions form a complete set of eigenfunctions.

Quantum Superposition and Interference

Quantum superposition is the principle that a particle can simultaneously exist in multiple states, and its overall wave function is a linear combination of the individual stationary state wave functions. When these states have different energies, their time-dependent phase factors can lead to interference effects, resulting in observable oscillations in measurable quantities such as the probability density.

Time Evolution of Wave Functions

The time evolution of a quantum state is governed by the Schrödinger equation, leading to the appearance of time-dependent phase factors in the wave functions of stationary states. These phase factors oscillate with a frequency related to the energy of each state. When different states are superposed, the relative phase evolution produces time-dependent interference effects in the overall wave function.

Probability Density and Oscillation Frequency

The probability density is obtained by taking the square modulus of the time-dependent wave function and reflects the likelihood of finding the particle at a given position and time. In superposed states, cross-terms appear due to interference between different stationary states, leading to oscillations in the probability density. The frequency of these oscillations is determined by the difference in energies of the constituent stationary states and is expressed in terms of angular frequency.

Transcript

0:00 Hello everyone.

00:01 In this problem, we are given the wave function to be the superposition of the first and third excited states of the way functions in a finite or sorry, an infinite square well of length l.

00:18 So we're told that the wave function looks like the following.

00:20 So we have one over root two times the spatial distribution of the ground state wave function, so upside 1x, times its corresponding time evolution function, which is ether to minus ie1 times t over hbar, plus 1 over 2 times the same thing for side 3, the third, or the second excited state.

00:44 Sorry, third excited state.

00:47 And then we're as to find the probability distribution.

00:50 The probability distribution is just the mot squared of this wave function, of this entire wave function, right? so it's si start, psi.

01:00 And so all we have to do is, is basically multiply this function by its complex conjugate.

01:05 And if you do that, then you should get something like a half, psi 1x squared, or mod squared, plus a half psi3, x, moth squared, plus 1 over 2.

01:17 So half, psi 1 star, psi 3 times e3, it over h4, times e1 and minus 3, plus a half times psi 1 times psi 3 star, ether than minus it over h4 times e1 minus e3.

01:29 Now the thing to notice here is that two of these parts, so two things in here are real, right? so the psi 1x mod squared is real and the psi 3 x mod squared is real.

01:42 And the other parts, so this function and this function are complex conjugates to be d.

01:50 So if you call this z and you call this z star, then you have z plus z star.

01:59 What is that equal to? well, if you have something like a complex number, like x plus iy, for example, and then you add to that its complex conjugate, then you're going to add plus x minus iy.

02:11 So you're going to get two times x.

02:14 So you're going to get two times the real part of your complex number.

02:19 So two times the real part of z is equal to 2x, right? so basically what you're going to end up with then is that the problem of distribution function is going to be equal to a half times psi 1 mod squared plus psi 3 mod squared plus 2 times the real part of this complex number that we got here.

02:41 Now we're going to use we're going to rely on the usual form of these wave functions, psi 1x and psi 3x.

02:48 So psi 1x is equal to the square rate of 2 over l, sine pi x over l, and this is equal to the complex conjugate of this spatial wave function because it's because it's a real function.

02:59 And psi 3x is equal to the square root of 2 over l times sine of 3 pi x over l.

03:05 And this is, of course, equal to its own compass conjugate as well.

03:09 So we can use this to find, you can use these expressions to find what the values are at l over 2, which is the position where we're interested in finding the problem and distribution function.

03:20 So if we put in l over 2 is equal to x, we find that psi 1 is equal to root 2, l and psi 3 is equal to minus over 2 over l...

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