A nonrelativistic electron is confined to a length of 500 pm on the x-axis. What is the kinetic energy of the electron if its speed is equal to the minimum uncertainty possible in its speed? `h =...

A nonrelativistic electron is confined to a length of 500 pm on the x-axis. What is the kinetic energy of the electron if its speed is equal to the minimum uncertainty possible in its speed? `h = 6.626 * 10^(-34) J * s,` `m_(el) = 9.11 * 10^-31 kg,` `1 eV = 1.60 * 10^(-19) J`). I got both .15 eV and 1.5 eV and I dunno if correct.

Choices are A) 1.5 eV B) .015 eV C) 150 eV D) 15 eV E) .15 eV

Wordz from old buddy previous test said "The minimum kinetic energy of the electron is closest to" not this minimum uncertainty stuff. old answer was .15 eV.

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator

Hello!

Again, we have a deal with the uncertainty principle for a position and a momentum. Mathematically it is expressed as

`Delta p*Delta x gt= h/(4 pi),`

where `Delta p = m*Delta v` is an uncertainty in a momentum and `Delta x` is the uncertainty in a position.

Also we know that the kinetic energy of an electron is equal to `(m_(el) v^2)/2,` and it is given that the speed is the same as the minimum uncertainty of a speed. So the kinetic energy is equal to

`E_k = (m_(el) v^2)/2 =(m_(el) (Delta v)^2)/2 = m_(el)/2 * (h/(4 pi) * 1/(m_(el)*Delta x))^2 =1/2 (h/(4 pi))^2 1/(m_(el)*(Delta x)^2).`

All values are given, and we can compute the result in Joules (note that p means pico- means `10^(-12)` ):

`E_k = 1/2*(1.055*10^(-34))^2/(9.11*10^(-31)*25*10^(-20)) = (1.055)^2/(2*9.11*25)*10^(-17) approx 0.00244*10^(-17) (J).`

To convert this to `eV` we have to divide by `eV` value:

`E_k =0.00244*10^(-17) / (1.60*10^(-19)) = 0.00244*10^2/1.60 approx0.153 (eV).`

So your hypothesis is correct, the answer is about `0.15 eV,` E.

weare878750 | Student

No actually 1.5 eV is correct I asked my professor but how do you get that then?

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