Phys 2601 Quiz #2 formulae
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Quiz #2 formulae
Phys 2601
1 Useful Formulae
Constants and unit conversions
h = 6.63 x 10 —34 Js = 4.14 x 10 — 15 eVs
方 = 
= 1.06 x 10 —34 Js = 6.58 x 10 — 16 eVs
c = 3.0 x 108 m/s
1 eV = 1.602 x 10 — 19 J
e = 1.60 x 10 — 19 C
me = 9.11 x 10 —31 kg = 0.511 MeV/c2
me /方2 = 1.31 x 1019 ( eV m2 ) — 1
kB = 1.38 x 10 —23 m2 kg s —2 K — 1 (Boltzmann’s constant) σ = 5.67 x 10 —8 W / (m2 K4 ) (Stefan’s constant)
λC = 
= 2.43 x 10 — 12 m (for an electron)
Useful formulae
General:
|
E = hν E = pc (m0 = 0) λdB = h/p ∆λ = λC (1 _ cos θ)
Intensity = ∆x∆px 2 2(方) κ = 1/λ
vp = ν/κ , vg =
En = _ ╱ Ze2 4π∈0、2 2方 |
KEmax = hν _ w0 E = )p2 c2 + m0(2)c4 p = mv/^1 _ v2 /c2 λ = c/ν (electromagnetic radiation) nλ = d sin θ
∆E∆t 2 2(方) |
(1)
(2)
Schroedinger Equation in 1 dimension:
pˆ = _i方∂/∂x
k = (2m(E _ V0 )/方 , κ = (2m(V0 _ E)/方
i方∂t Ψ(x, t) = _2m(方2)∂x(2)Ψ(x, t) + V (x)Ψ(x, t)
Eψ(x) = _2m(方2)∂x(2)ψ(x) + V (x)ψ(x)
)B) 2 v2 )C) 2
T = '(┌)1 + 
— 1 s 16 
╱ 1 _ 
、 e —2κ2 a
T = ┌ 1 + sin2 k2 a ┐ — 1
' 4 
╱ 
_ 1、|
Useful integrals
!—-(-) e —ax2 dx = ( 
!—-(-) x2n+1e —ax2 dx = 0 any integer n
!—-(-) x2 e —ax2 dx = 
( 
!—-(-) x4 e —ax2 dx = 
( 
(3)
(4)
2 Short answer problems
3 points each. Answer each of the following problems with a sentence, or a brief (1-line) calculation.
1. Is the state ψ(x) = sin(kx) = 
(eikx _ e —ikx ) an eigenstate of the momentum operator?
2. An electron makes a transition from the n = 3 state of hydrogen to the ground state. What is the frequency of the emitted photon?
3. Normalize the wave-function ψ(x) = e —x2 /(2b2 ) . (The table of integrals in the formula sheet may be useful).
4. Calculate op2 ( for the wave function ψ(x) = sin (nπx/a) in the region 0 s x s a, with ψ(x) = 0 elsewhere.
5. An electron with energy 2 eV is incident on a barrier of height 3 eV and a thickness of 10 —9 m. What is the transmission amplitude for this particle to tunnel through the barrier?
6. Consider the potential V (x) = ^)x), sketched below. Does the Schroedinger equation for this potential have any solutions that are not bound states?
3 Long Answer problems
16 points each. Please answer the following questions. Show your work!
1. Consider the wave function
ψ(x) =^2 ╱ 
、1/4 xe —αx2 /2 , α = mω/方 (5)
(a) Does this wave function satisfy the time-independent Schroedinger equation for V (x) = 
mω 2 x2 ? Be sure to show a calculation justifying your result.
(b) What is the expectation value of the energy operator Hˆ = _方2 /(2m)∂x(2) + 
mω 2 x2 in the state ψ(x)? Again, be sure to show a calculation establishing your result. Note: the state is already normalized.
(c) Compute op2 ( _ op(2 for this state. Is it an eigenstate of momentum? Hint: there is a table of useful integrals on the formula sheet!.
2. Consider the potential
V (x) =
V1
x < 0
x > 0
(6)
(a) For each of the two regions (I: x < _0; II: x > 0) write the general solution to the time- independent Schroedinger equation when E < V1 . Include explicit expressions for the constants kI and κII .
(b) Write three equations that the free parameters associated with this solution must satisfy. Be sure to include any equations that are needed to ensure that the wave function is finite everywhere.
(c) What is the reflection coefficient in this case? You may answer this using an explicit calculation, or by giving a physical argument.
(d) Suppose you begin to increase E . For what value of E will the reflection constant from (c) change? Do NOT do a calculation to answer this. Instead, explain the difference based on the probability flux far from the barrier in both cases. Hint: Consider what happens to ψ(x) as x → o . How does that relate to the probabilty of the particle being on either side of the barrier?
2023-03-23