Question

The wave function for a particle moving along the positive x-direction is given by si(x,t)=Ae^t(kx-wt).Use this obtain an expression for the momentum and KE operator in 1d.​

Answer 1

Answer:

Explanation:

In classical mechanics the state of motion of a particle is specified by the particle’s

position and velocity. In quantum mechanics the state of motion of a particle is given

by the wave function. The goal is to predict how the state of motion will evolve as time

goes by. This is what the equation of motion does. The classical equation of motion

is Newton’s second law F = ma. In quantum mechanics the equation of motion is the

time-dependent Schroedinger equation. If we know a particles wave function at t = 0, the

time-dependent Schroedinger equation determines the wave function at any other time.

The states of interest are the ones where the system has a definite total energy. In

these cases, the wave function is a standing wave. When the time-dependent Schroedinger

equation is applied to these standing waves, it reduces to the simpler time-independent

Schroedinger equation. We will use the time-independent Schroedinger equation to find

the wave function of the standing waves and the corresponding energies. So when we say

“Schroedinger equation”, we will mean the time-independent Schroedinger equation.

Even though the world is 3 dimensional, let’s start by considering the the simple

problem of a particle confined to move in just one dimension. For example, imagine an

electron moving along a very narrow wire.

Classical Standing Waves

Let’s review what we know about classical standing waves in 1D. Think of waves on

a string where the string’s displacement is described by y(x, t). Or we might consider a

sound wave with a pressure variation p(x, t). For an EM wave, the wave function of the

electric field would be E~ (x, t). We’ll consider waves on a string for concreteness, but this

will apply to all kinds of 1D waves, so we’ll use the general notation Ψ(x, t) to represent

the wave function.

Let us consider first 2 sinusoidal traveling waves, one moving to the right,

Ψ1(x, t) = B sin(kx − ωt) (1)

and the other moving to the left with the same amplitude

Ψ2(x, t) = B sin(kx + ωt) (2)

The superposition principle guarantees that the sum of these two waves is itself a possible

wave motion:

Ψ(x, t) = Ψ1(x, t) + Ψ2(x, t) = B[sin(kx − ωt) + sin(kx + ωt)] (3)

Using the trigonometric identity

sin a + sin b = 2 sin

a + b

2

!

cos

a − b

2

!

(4)

we can rewrite Eq. (3)

Ψ(x, t) = 2B sin kx cos ωt (5

Answer 2

Concept

The location and velocity of a particle define its state of motion in classical mechanics.

The wave function in quantum physics determines a particle's state of motion. Predicting how the condition of motion will change over time is the ai

The second law of Newton, F = ma, is the standard equation for motion.

The time-dependent Schrodinger equation serves as the fundamental equation of motion in quantum physics.

The time-dependent Schrodinger equation may be used to calculate a particle's wave function at any other time if we know it at t = 0

Given

the positive x-direction is  si(x,t)=Ae^t(kx-wt).

Find

We need to find the momentum and Kinetic energy

Solution

Let's have a look at the first two sinusoidal travelling waves.

Ψ1(x, t) = B sin(kx − ωt) (1)

and the other moving to the left with the same amplitude

Ψ2(x, t) = B sin(kx + ωt) (2)

The Wave motion is :

Ψ(x, t) = Ψ1(x, t) + Ψ2(x, t) = B[sin(kx − ωt) + sin(kx + ωt)] (3)

Using the trigonometric identity

$sin a + sin b = 2sin \frac{ a+b}{2} cos\frac{a-b}{2}$

Rewriting the equation

Ψ(x, t) = 2B sin kx cos ωt

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