Quantum mechanics - several constant potentials

In summary: I would suggest that you say something like "I'm using Å⁻¹ for α".The ångström (Å) (note spelling and capitalisation) is a bit old-fashioned, but is still used in some areas. It is not an SI unit.1 Å = 10⁻¹⁰ mThat means1 Å⁻¹ =10¹⁰ m⁻¹I would have given the answer as:α = 8.76x10⁸ (m⁻¹ assuming we are working in SI units)EDIT: It's very poor practice to give values without units. The question should have given the units for k and b. I would suggest
  • #1
Eitan Levy
259
11
Homework Statement
A particle with mass m is in a one dimensional potential, as seen below.

The wave function in [itex]x<0[/itex] is 0.

The wave function in [itex]0<x<b[/itex] is: [itex]Asin(kx)[/itex]

The wave function in [itex]x>b[/itex] is: [itex]Be^{-\alpha x}[/itex]

It is known that [itex]k=3*10^{10}[/itex] and [itex]b=0.5333333*10^{-10}[/itex]

Find [itex]\alpha[/itex]
Relevant Equations
Schrodinger stationary equation
Capture.PNG


What I tried to do was using the fact that the wave function should be continuous.

[itex]Asin(kb)=Be^{-\alpha b}[/itex]

The derivative also should be continuous:

[itex]kAcos(kb)=-\alpha Be^{-\alpha b}[/itex]

And the probability to find the particle in total should be 1:

[itex]\int_0^b A^2sin^2(kx) dx + \int_b^{\infty} B^2e^{-2\alpha x} dx =1 [/itex]

This set of equations is to hard to deal with, the equations should be solved with calculator only so I think I did something wrong.

Also, there may be a better way to approach this problem, but I'm not seeing it.
 
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  • #2
Eitan Levy said:
[itex]Asin(kb)=Be^{-\alpha b}[/itex]

[itex]kAcos(kb)=-\alpha Be^{-\alpha b}[/itex]

[itex]\int_0^b A^2sin^2(kx) dx + \int_b^{\infty} B^2e^{-2\alpha x} dx =1 [/itex]
Hints: It's much easier than you think. You only need the first 2 equations.
 
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  • #3
Steve4Physics said:
Hints: It's much easier than you think. You only need the first 2 equations.
Is the answer 0.08763*10^(10)?
 
  • #4
Eitan Levy said:
Is the answer 0.08763*10^(10)?
I haven't done the calculation. How did you get that value?
Also, have you forgotten units or were there really none supplied in the original question?
 
  • #5
Steve4Physics said:
I haven't done the calculation. How did you get that value?
Also, have you forgotten units or were there really none supplied in the original question?
I divided the first equation by the second:

[itex]\frac{1}{k}tan(kb)=-\frac{1}{\alpha}[/itex]

And I plugged in the numbers.

No units were provided.
 
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  • #6
Eitan Levy said:
I divided the first equation by the second:

[itex]\frac{1}{k}tan(kb)=-\frac{1}{\alpha}[/itex]

And I plugged in the numbers.

No units were provided.
Agreed. Well done! A couple of points:
1) You haven't correctly used standard form (mantissa in wrong range).
2) You have rounded the last digit incorrectly when giving the answer to 4 significant figures.

If working in SI units, what do you think the unit would be? (If I were handing this in, I would make a comment about the appropriate SI units below my answer.)
 
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  • #7
Steve4Physics said:
Agreed. Well done! A couple of points:
1) You haven't correctly used standard form (mantissa in wrong range).
2) You have rounded the last digit incorrectly when giving the answer to 4 significant figures.

If working in SI units, what do you think the unit would be? (If I were handing this in, I would make a comment about the appropriate SI units below my answer.)
[itex](Angstram)^{-1}[/itex]
 
  • #8
Eitan Levy said:
[itex](Angstram)^{-1}[/itex]
The ångström (Å) (note spelling and capitalisation) is a bit old-fashioned, but is still used in some areas. It is not an SI unit.

1 Å = 10⁻¹⁰ m
That means
1 Å⁻¹ =10¹⁰ m⁻¹

I would have given the answer as:
α = 8.76x10⁸ (m⁻¹ assuming we are working in SI units)

EDIT: It's very poor practice to give values without units. The question should have given the units for k and b.
 

Related to Quantum mechanics - several constant potentials

1. What is quantum mechanics?

Quantum mechanics is a branch of physics that studies the behavior of particles at a very small scale, such as atoms and subatomic particles. It describes how these particles interact with each other and with energy, and has been successful in explaining many phenomena that classical physics could not.

2. What are constant potentials in quantum mechanics?

In quantum mechanics, a potential is a mathematical function that describes the force acting on a particle. A constant potential is one that does not change with time or position. It is often used to represent a uniform electric or magnetic field.

3. How do constant potentials affect the behavior of particles?

Constant potentials can affect the behavior of particles in different ways, depending on the specific potential. For example, a constant electric potential can cause a charged particle to accelerate, while a constant magnetic potential can cause it to move in a circular path. In general, constant potentials can alter the energy levels and trajectories of particles.

4. What are some examples of constant potentials in quantum mechanics?

Some examples of constant potentials in quantum mechanics include the infinite square well potential, the harmonic oscillator potential, and the hydrogen atom potential. These potentials are commonly used to model different physical systems and have been studied extensively in quantum mechanics.

5. What is the significance of studying quantum mechanics with constant potentials?

Studying quantum mechanics with constant potentials allows us to better understand the fundamental principles of quantum mechanics and how they apply to different systems. It also helps us develop mathematical tools and techniques for solving more complex problems, and has practical applications in fields such as nanotechnology and quantum computing.

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