Chapter 4: Q4P (page 139)

Show that $Θ = A I n [t a n (\frac{θ}{2})]$ satisfies the $θ$ equation (Equation 4.25), for l = m = 0. This is the unacceptable "second solution" -- whats wrong with it?

Short Answer

Expert verified

$Θ (θ) = A I n [\tan \frac{θ}{2}]$ does satisfy the equation but this is the unacceptable second solution since $Θ$ blows up at $θ = 0$ and at $θ = π$

Step by step solution

Define the Schrödinger equation

A differential equation describes matter in quantum mechanics in terms of the wave-like properties of particles in a field. Its answer is related to a particle's probability density in space and time.

Calculation

We need to show that,

$Θ (θ) = A I n [\tan \frac{θ}{2}]$

Satisfies the equation

$\sin (θ) \frac{d}{d θ} (\sin (θ) \frac{d Θ}{d θ} + [I (I + 1) \sin^{2} (θ) - m^{2}] Θ = 0$

So now take the derivative then we get,

$\frac{d Θ}{d θ} = \frac{A}{\tan (\frac{θ}{2})} \frac{1}{2} \sec^{2} (\frac{θ}{2}) \frac{d Θ}{d θ} = \frac{A}{2} \frac{1}{\sin (\frac{θ}{2}) \cos (\frac{θ}{2})} = \frac{A}{\sin θ}$

Therefore,

$\frac{d}{d θ} (\sin θ \frac{d Θ}{d θ}) = \frac{d}{d θ} A = 0$

Then, we get,

role="math" localid="1656064432888" $Θ (0) = A I n (0) = A (- \infty) Θ (π) = A I n (\tan \frac{π}{2}) = A I n (- \infty) = A (\infty)$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Show that $Θ = A I n [t a n (\frac{θ}{2})]$ satisfies the $θ$ equation (Equation 4.25), for l = m = 0. This is the unacceptable "second solution" -- whats wrong with it?

Short Answer

Step by step solution

Define the Schrödinger equation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Dynamics

Measurements

Kinematics Physics

Linear Momentum

Circular Motion and Gravitation

Study anywhere. Anytime. Across all devices.

Show thatΘ=AIn[tan(θ2)]satisfies the θequation (Equation 4.25), for l = m = 0. This is the unacceptable "second solution" -- whats wrong with it?

Short Answer

Step by step solution

Define the Schrödinger equation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Dynamics

Measurements

Kinematics Physics

Linear Momentum

Circular Motion and Gravitation

Study anywhere. Anytime. Across all devices.

Show that $Θ = A I n [t a n (\frac{θ}{2})]$ satisfies the $θ$ equation (Equation 4.25), for l = m = 0. This is the unacceptable "second solution" -- whats wrong with it?