Quantum Mechanics: Speed of an electron

[hquiquero]

Homework Statement

Find the energy (in joules), rest mass (kg), speed (m/s), wavelength (m), and momentum (kgm/s) of a 2.2 eV electron.

I'm having trouble finding the speed of the electron; once I have that I can find the wavelength and momentum.

E = energy
p = momentum
h = Planck's constant (6.63 x 10^-34 Js)
c = the speed of light (3.0 x 10⁸ m/s)
λ = wavelength

Homework Equations

1 eV = 1.60 x 10^-19 J

E_total = E_k + E_rest

E = hc/λ

p = h/λ

The Attempt at a Solution

Energy of electron in joules:
2.2 eV = 3.52 x 10^-19 J

Rest mass of an electron:
9.11 x 10^-31 kg
****(Is this a correct answer or do I need to calculate rest mass?)

Speed of electron:

I tried this equation:
E_total = E_k + E_rest
E_k = E_total - E_rest

but it doesn't work because the total energy given for the electron is less than the rest energy calculated...
I'm not sure what equation other equation I can use. Also, I don't know if relativity must be taken into account if the electron's speed is close to the speed of light?

Thank you!

 

[BruceW]

E = hc/λ

This is not true, because frequency is not equal to c/λ. You should leave it as E=hf.

 

[BruceW]

Speed of electron:

I tried this equation:
E_total = E_k + E_rest
E_k = E_total - E_rest

but it doesn't work because the total energy given for the electron is less than the rest energy calculated...
I'm not sure what equation other equation I can use. Also, I don't know if relativity must be taken into account if the electron's speed is close to the speed of light?

Thank you!

The equation is correct, but what are you using for their values? The question gives you E_k, and you should be trying to find E_total.

Also, you've come across an important point when you said if the electron's speed is close to the speed of light. Compare the rest energy to the E_k and you'll get the answer to this.

 

[hquiquero]

This is not true, because frequency is not equal to c/λ. You should leave it as E=hf.

Sorry, that equation was for another part of the question about photons that I decided to leave out... the equation I'd use to find wavelength of the electron is λ = h/mv

 

[hquiquero]

The equation is correct, but what are you using for their values? The question gives you E_k, and you should be trying to find E_total.

Also, you've come across an important point when you said if the electron's speed is close to the speed of light. Compare the rest energy to the E_k and you'll get the answer to this.

I was using the given value for energy for the total energy, but when I calculated the rest energy you get

E_rest = mc²
=(9.11 x 10^-31)(3.0 x 10⁸)²
=8.199 x 10^-14 J

which is more than the given energy...

Am I supposed to use the given energy for the kinetic energy? And if so, does it just equal to 1/2mv²?

 

[BruceW]

the energy they give must be the kinetic energy, because it is less than the rest energy.

In relativity, the KE is not 1/2mv^2. I think you should look over the basic definitions again. are you learning from a textbook, or a teacher or online? Wikipedia give a fairly good explanation.

What are you trying to calculate in this question? speed and relativistic momentum?

 

[hquiquero]

I am learning from an online course... I tried using the E_k = 1/2mv² and got a speed that was only 0.3% of the speed of light, so I think that means I don't have to consider relativity in my calculations.. I could be wrong though?

 

[BruceW]

You are correct that the speed of this particle is much less than the speed of light. But the equation you are using (Ek=1/2mv²) is not correct. It is only approximate when the speed of the particle is much less than the speed of light, and it is totally untrue when the particle's speed is close to the speed of light.

Do you know the equation for the energy of a particle in terms of gamma? This is where you should start to find the true speed of the particle.

Or you could just say that since the particle's speed is small compared to the speed of light, you could just treat it as a classical particle. (But this would be an approximate answer).