(III) (a) Show that Ψ(x, t)=A e^i(k x-a t) is a solution to the time-dependent Schrodinger equat

(III) (a) Show that Ψ(x, t)=A e^i(k x-a t) is a solution to...

Key Concepts

de Broglie Relations

The de Broglie relations form a bridge between particle and wave properties by associating the wavelength with momentum and the frequency with energy. These relations are essential in connecting classical and quantum descriptions of matter, allowing quantities such as momentum and energy to be expressed in terms of wave parameters. This concept is foundational in establishing the dispersion relations and validating the solution of the Schrödinger equation for free particles.

Time-dependent Schrödinger Equation

This is a fundamental equation in quantum mechanics that governs the evolution of a quantum state in time. It is a partial differential equation involving the Hamiltonian operator and describes how the wave function changes under the influence of kinetic and potential energy. The equation ensures that the probabilistic interpretation of quantum mechanics is preserved through its linear and unitary time evolution.

Free Particle

In quantum mechanics, a free particle is one that is not subjected to any external forces, meaning its potential energy is constant across space. The free particle scenario simplifies the Schrödinger equation by removing spatial dependence from the potential term, thereby focusing on the intrinsic kinetic behavior of a quantum system. It forms the basis for understanding more complex quantum systems.

Complex Exponential Solutions

Complex exponential functions, such as those of the form exp[i(kx ? ?t)], are natural solutions to the time-dependent Schrödinger equation because of their favorable differentiation properties. These functions are eigenfunctions of the momentum operator and facilitate the expression of particle-like behavior via a well-defined phase, ensuring that quantum mechanical operators act in a linear and consistent manner.

Real Sinusoidal Wave Functions

Although sine and cosine functions are mathematically related to complex exponentials, they do not independently satisfy the Schrödinger equation in its differential operator form. This is because the Schrödinger equation inherently involves complex potentials and provides a framework where phase information is crucial, making the complex exponential representation indispensable in capturing the full quantum behavior.

Conservation of Energy in Quantum Mechanics

Conservation of energy is a core tenet in all physical theories, including quantum mechanics. In the context of the Schrödinger equation, it implies that the total energy, represented by the Hamiltonian, must remain constant over time. For a solution to be physically viable, it must respect this conservation law, which, in the case of a free particle, is demonstrated when the de Broglie relations are used to relate the frequency and wave number with the energy and momentum.

Transcript

00:01 In the first item, we have to calculate and check if psi 1, 2 and 3 are solutions of the time -dependent throughout the equation or not.

00:09 We begin by checking psi 1.

00:14 What is the time derivative of psi 1? v -psi over d -t.

00:20 This is equal to a, which is a constant.

00:24 The derivative of the exponential is itself times the derivative of the argument of the exponential.

00:35 So it multiplies the time derivative of i times k times x minus a times t.

00:44 But note that the first term k times x do not have a t dependence.

00:49 So it will not contribute to that time derivative.

00:53 So we will only have minus i times a times the derivative of t with respect to t, which is equals to 1.

01:03 Then this is equal to minus i times a times x again.

01:11 Now what is the derivative of psi 1 with respect to x? this is equal to a again times the exponential because the derivative of the exponential itself times the derivative of the exponential with respect to x.

01:31 The derivative of the argument of the exponential with respect to x is equal to i times k times the derivative of x with respect to x, which is one.

01:43 This term will not give a contribution since it doesn't depend on x.

01:48 Therefore, we have this times i times k, and that's it.

01:54 So this is equal to i times k times psi 1.

01:58 So, as we can see, the time derivative is the same thing as multiplying by minus i times a, while the spatial derivative is the same thing as multiplying by i times k.

02:11 Therefore, we can plug in these results in the time -dependent runningers equation to check if it is satisfied it.

02:18 So we have minus h -bar squared divided by 2m times.

02:24 We will apply the spatial derivative two times.

02:29 So we have minus, not minus, we have i times k squared times xi.

02:37 Because by applying the derivative once, we get a factor of i times k.

02:41 And by applying the derivative again, we get another factor of i times k, which amounts two of i times k squared.

02:49 Plus alpha, which is the constant value for the potential times psi one.

02:58 And this is equal to should be i times age times the time derivative of psi 1 then we have minus i times z a times psi 1 now we can elaborate this expression that we have minus 8 bar squared times k squared note that we have an i squared here which gives an extra factor of minus 1 canceling out this same signal.

03:30 This is divided by 2 times m and multiplies psi 1 plus alpha times psi 1 is equal to h times h bar times a times psi 1.

03:50 Therefore we have h bar squared times k squared divided by 2m plus alpha times psi 1 being equals to age, bar times a times psi 1 so is this equation satisfied well yes it is if and only if the following condition is also satisfied so if this condition is satisfied then x1 is a solution now let us check px2 for psy 2 the time derivative is given by the following d psi 2 d t is equal equals to a times the derivative of the cosine.

04:51 The derivative of the cosine is minus sine, k times x minus omega times t.

05:00 Now we have to multiply it by the derivative of the argument of that cosine.

05:06 The derivative with respect to t, then we get a factor of minus omega coming from that derivative.

05:14 And this is equal to well it is equals to minus with minus plus, so a times omega times the sign of k times x minus omega times t.

05:31 This is the time derivative of side 2 with respect to time.

05:36 Now for the spatial derivative we have the following.

05:43 The spatial derivative is again very similar a times minus sign of k times x minus x minus omega times t, times the derivative of the argument of that sign with respect to x, and this gives us an extra factor of k.

06:03 So we have this times k.

06:06 And then this is equal to a times k times k times of k times x minus omega times t, with a minus sign in fronten you're ahead of it...

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