A particle of mass m in the one-dimensional potential energy...

Key Concepts

Quantum Measurement and Probabilities

In quantum mechanics, the act of measurement collapses a state into one of the eigenstates of the observable being measured, with probabilities given by the squared magnitudes of the coefficients in the state’s expansion in that eigenbasis. This key tenet underpins the probabilistic interpretation of quantum mechanics and explains how specific outcomes are obtained from superpositions.

Time Dependence of Observables

Even if a quantum state is built as a superposition of stationary states, observables that do not commute with the Hamiltonian can have time-dependent expectation values. This occurs because different eigenstates acquire different phase factors over time, leading to interference effects that make quantities such as the average position vary with time.

Expectation Value in Quantum Mechanics

The expectation value represents the average outcome of a measurement over many identical experiments performed on a quantum system. It is computed by integrating the product of the state, the observable, and the complex conjugate of the state. Expectation values provide a bridge between the probabilistic nature of quantum mechanics and classical measurable quantities.

Superposition Principle

The superposition principle allows a quantum state to be expressed as a linear combination of eigenstates. Each eigenstate is weighted by a complex coefficient, and the full state evolves by the time-dependent phases associated with each eigenstate. This principle is central to understanding dynamics and interference effects in quantum systems.

Stationary States

Stationary states are the energy eigenstates of a quantum system; they are solutions to the time-independent Schrödinger equation with definite energies. In these states, the probability density remains constant over time, and they form an orthonormal set that can be used to construct more general states by superposition.

Infinite Potential Well

This is a fundamental quantum system where the potential energy is zero (or constant) inside a finite region and infinite outside, effectively confining the particle to that region. The impenetrable boundaries lead to quantized energy levels and eigenfunctions that satisfy specific boundary conditions, which form the basis for many problems in quantum mechanics.

Time Evolution of Quantum States

Time evolution in quantum mechanics is governed by the time-dependent Schrödinger equation. For systems with a known energy eigenbasis, each eigenstate evolves with a characteristic phase factor exp(-iE_n t/?). This formalism enables the determination of the state at any later time from its initial configuration.

Transcript

00:01 In this problem you're given an infinite well and you're told the wave function at t equals zero is this quantity here and that's inside the well and zero elsewhere.

00:12 And it asks us some questions.

00:14 The first being what is the wave function as a function of time in addition.

00:20 So let's look at that now, part a.

00:23 Now the time dependent, shown in this equation, that involves the expression that comes from that, the time aspect, involves the energy.

00:36 So we need to look into this to see what are the, how can we write this in terms of energy eigenstates.

00:44 That's what we need as you're going to see.

00:47 So let me write out those eigenstates.

00:50 I'll use a phi to identify those.

00:54 So these are the energy eigenstates.

01:03 And this is equal to the square root of two over l, sine n pi x over l.

01:13 The energy associated with each of those is n squared pi squared h bar squared over 2ml squared.

01:23 And n runs from one and two and beyond.

01:30 So they made this very easy for us.

01:32 We can easily see that this, sometimes it's hard to actually see.

01:38 It's such a complicated thing that you actually have to go through the proper procedure, mathematical procedure to get the actual, what actually is the expansion into the eigenstates.

01:52 What are the coefficients really of each of those eigenstates.

01:56 But here it's very easy.

01:58 This will be phi one, phi two.

02:03 So psi of x, one plus i over two, phi one plus one over the square root of two, phi two.

02:15 There we have it.

02:16 So we now have it in terms of energy eigenstates.

02:20 Now the idea, the superposition that takes place, this takes, is the same at all times.

02:29 So really you can almost reverse the process.

02:31 You can think of psi x t as superposition of the time dependent energy eigenstate for one, energy e1, and the time dependent for e2.

02:45 And then it just happens to work at t equals zero, then certainly that'll be true also.

02:51 So that's how you think about it.

02:53 So we really need the solution of the time dependent schrodinger equation in here.

03:00 And in here, if i want the time dependent wave function.

03:04 So let me write that out.

03:06 Psi of x t, one plus i over two, psi one of x, exponential.

03:14 This is the time part that would have come from the time dependent schrodinger equation.

03:29 And then we do it for the other term.

03:40 Okay, there we have it.

03:41 So again, this whole thing is the solution of the time dependent schrodinger equation.

03:49 That's how it works.

03:51 So that's how we think of the superpositions.

03:54 And let me write out what the energies are, the formula for that.

03:58 E1 pi squared h bar squared, or two m l squared.

04:06 E2, just four e1.

04:09 If you remember, you look at the formula, four e1.

04:12 So four pi squared h bar squared, two m l squared.

04:18 And this just can be reduced to two pi squared h bar squared m l squared.

04:30 So there's our two energies.

04:32 Now we can write out the wave function with that included.

04:36 I'm not gonna put in the sign, the original pieces.

04:40 We don't need them.

04:41 We're never gonna be integrating over x to actually need that.

04:45 But we do need to look at the time aspect.

04:47 So i'll write that out explicitly.

04:56 So just putting in e1 minus i pi squared h bar t over two m l squared.

05:03 Remember there is an h bar already.

05:06 So the h bar squared becomes h bar plus one over square root of two, phi two x exponential minus i two pi squared h bar t over m l squared.

05:23 There we have it.

05:24 So there is our wave function as a function of time.

05:30 And certainly, it works at t equals zero, also it has to.

05:38 Okay, that was part a.

05:42 Part b wants to know the expectation value of the energy.

05:46 And that's found by multiplying the probability of e1 times e1 plus the probability of e2 times e2.

06:00 So you might say, this is something unusual, not really.

06:04 How do you get probability here? got 10 ,000 measurements, 6 ,000 of them are e1, 4 ,000 are e2.

06:13 The probability for e1 is 6 ,000 divided by 10 ,000 .6.

06:17 Probability for e2 is 4 ,000 divided by 10 ,000 .4.

06:22 Notice it adds to one, which it has to because you're only gonna get e1 or e2.

06:25 You're not gonna get anything else.

06:28 Also, that 0 .4 can also be thought of as one minus the probability of e2 is one minus the probability of e1, same thing.

06:39 So all depends.

06:43 Now, how do we get the probability? let me write it out, then i'll explain it.

06:58 Let's look first at the integral.

07:00 This is the inner product, scalar product, dot product.

07:04 It tells me the phi1 component of psi, just like i dot f gives me the x component of f.

07:15 Exactly the same thing...

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