The Simple Harmonic Oscillator (again)

Question 1

What is the equation for the potential V of a SHO?

Answer 1

V = 1/2k’x^2, where k’ = m*ω^2

Question 2

What is the equation for the energy En of a SHO?

Answer 2

En = (n+1/2)ћω

Question 3

What is the equation for the ground state eigenfunction u0 for the SHO?

Answer 3

u0 = (mω/πћ)^1/4 *exp(-(mωx^2)/2ћ)

Question 4

How can we find , , <p> and </p><p>?</p>

Answer 4

Use = integral from -inf to inf of u* Q(hat)u dx -> by symmetry, =0 and <p> = 0 on the SHO graph</p>

Question 5

What do we get for the equation of ?

Answer 5

= integral from -inf to inf of u0x^2u0 dx = (mω/πћ)^1/2 *(π^1/2 *ћ^3/2)/(2m^3/2 *ω^3/2) = ћ/2mω

Question 6

What do we get for the equation of <p>?</p>

Answer 6

<p> = mωћ/2</p>

Question 7

What happens if we plug our results for and <p> into the generalised HUP formula?</p>

Answer 7

Δx = sqrt(-^2) = sqrt() = sqrt(ћ/2mω), and Δp(x) = sqrt(mωћ/2) -> multiply these together and we get ΔxΔp(x) = ћ/2 >= ћ/2, as required by HUP

Question 8

What is u0(x)?

Answer 8

A minimum uncertainty state.

Question 9

How can we calculate E0 with operator formation?

Answer 9

= integral from -inf to inf of u0* H(hat)u0 dx, where H(hat) = p(x)(hat)^2/2m + 1/2k’x(hat)^2 -> rearrange and get E0 = ћω/2

Question 10

How can we factorise H(hat) and transform it into dimensionless variables?

Answer 10

H(hat) = -ћ^2/2m *d^2/dx^2 +mω^2/2 *x^2 -> transform to dimensionless variables by adding in g = sqrt(mω/ћ) *x and ε = 1/ћω *E, which are the reduced length and reduced energy respectively.

Question 11

What is the factorised version of H(hat)?

Answer 11

H(hat) = 1/2*(g^2 - d^2/dg^2)

Question 12

What is the equation for the raising operator?

Answer 12

a(+hat) = 1/sqrt(2) *(g-d/dg)

Question 13

What is the equation for the lowering operator?

Answer 13

a(-hat) = 1/sqrt(2) *(g+d/dg)

Question 14

What does [g, d/dg] equal?

Answer 14

Equals -1

Question 15

What do we get if we multiply the raising and lowering operators together in each direction?

Answer 15

a(+hat)*a(-hat) = H(hat) - 1/2, other direction gives H(hat) + 1/2

Question 16

How can we use the results for multiplying the raising and lowering operators together?

Answer 16

Use to find [a(+hat), a(-hat)] and [a(+hat), H(hat)] by subbing it in to the equation and rearranging.

Question 17

What does [a(+hat), a(-hat)] equal?

Answer 17

-1

Question 18

How do we operate with [a(+hat), H(hat)] on u(n)?

Answer 18

Multiply by u(n) and expand out the equation, then rearrange for H(hat)*u(n)’

Question 19

What do we get for H(hat)*u(n)’?

Answer 19

H(hat)u(n)’ = (ε(n) + 1)u(n)’, where u(n)’ is the eigenfunction and ε(n)+1 is the eigenvalue

Question 20

What does the raising and lowering operator take ε and u(n) to?

Answer 20

ε -> ε+1 –> E -> E+ћω and u(n) -> u(n)’. Lowering takes E->E-ћω

Question 21

What must there be a bottom ring for?

Answer 21

For which a(-hat)u0(g) = 0, so get H(hat)u0 = 1/2*u0

Question 22

What is the final result after all of this?

Answer 22

ε0 = 1/2, so E0 = ћω/2 -> expected