Q1. Consider the non-normalized waves packet for a particle in one-dimension is given as following: a S8N 41(x) = otherwise 2(x) = 8Ne-x-x,)²/a² eikox (a) Calculate (x). (P). (x²). (P²). Q2. If you are given the following operator 03, acts on function of position Þ(x). as: 034(x) = p(x) əx Or (translate-by-T) operator, acts on function of position Þ(x). as: 0rÞ(x) : Þ(x – T) (a) Show that: [0r.x] = – TOT (b) Show that: [0r.03] = 0
Q1. Consider the non-normalized waves packet for a particle in one-dimension is given as following: a S8N 41(x) = otherwise 2(x) = 8Ne-x-x,)²/a² eikox (a) Calculate (x). (P). (x²). (P²). Q2. If you are given the following operator 03, acts on function of position Þ(x). as: 034(x) = p(x) əx Or (translate-by-T) operator, acts on function of position Þ(x). as: 0rÞ(x) : Þ(x – T) (a) Show that: [0r.x] = – TOT (b) Show that: [0r.03] = 0
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![MOBILY RIYADH SEASON
4G
3G
V:0E
1Y,VK/s
VoWiFi STC KSA RIYADH SEASON
HOMEWORK 1
Q1. Consider the non-normalized waves packet for a particle in one-dimension is
given as
following:
otherwise
2(x) = 8Ne-(x-x,)²/a² eikox
(a) Calculate (x). (P). (x²). (P²).
Q2. If you are given the following operator
03, acts on function of position Þ(x). as: 034(x)
Þ(x)
ax
0, (translate-by-T) operator, acts on function of position Þ(x). as: 0rµ(x) =
Þ(x – T)
(a) Show that: [0r.x] = – TOT
(b) Show that: [0r.03] = 0
Q3. Consider an infinite potential well of width d. In transitions between neighboring
values of
n, particles of mass that is in a position state as:
1
+
2πχ
1
f(x.t) =
TTX
sinTe
-iwot
sin
e
-ίω1t
d
d
(a) Proof that ƒ(x.t) is still normalized for all value of t.
(b) Find the probability distribution P(x.t) = |f(x.t)|²
Q4. Proof that the probability current for a wavefunction of the form](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7408245c-a53a-45eb-810d-d52759d7ab58%2Faaf3729d-fa04-493d-b327-97e8860325b8%2Fvat98nh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:MOBILY RIYADH SEASON
4G
3G
V:0E
1Y,VK/s
VoWiFi STC KSA RIYADH SEASON
HOMEWORK 1
Q1. Consider the non-normalized waves packet for a particle in one-dimension is
given as
following:
otherwise
2(x) = 8Ne-(x-x,)²/a² eikox
(a) Calculate (x). (P). (x²). (P²).
Q2. If you are given the following operator
03, acts on function of position Þ(x). as: 034(x)
Þ(x)
ax
0, (translate-by-T) operator, acts on function of position Þ(x). as: 0rµ(x) =
Þ(x – T)
(a) Show that: [0r.x] = – TOT
(b) Show that: [0r.03] = 0
Q3. Consider an infinite potential well of width d. In transitions between neighboring
values of
n, particles of mass that is in a position state as:
1
+
2πχ
1
f(x.t) =
TTX
sinTe
-iwot
sin
e
-ίω1t
d
d
(a) Proof that ƒ(x.t) is still normalized for all value of t.
(b) Find the probability distribution P(x.t) = |f(x.t)|²
Q4. Proof that the probability current for a wavefunction of the form
![MOBILY RIYADH SEASON
4G
3G
V:0E
1,1•K/s
VoWiFi STC KSA RIYADH SEASON
Q4. Proof that the probability current for a wavefunction of the form
Þ(x) = R(x) eis(x)
ħ əs(x)
(a) j = |R(x)|²;
m
Əx
ħk
(b) j =
|R|² for a wavevector k.
т
Q5. Proof the following:
(a) Using the Cartesian components of L, Proof that: [Ly . Lz]y = 0
%3|
(b) Lž + L² # (Ly + iL,)(Ly – iL,)
(c) [L². L+] = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7408245c-a53a-45eb-810d-d52759d7ab58%2Faaf3729d-fa04-493d-b327-97e8860325b8%2Fwdkxse9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:MOBILY RIYADH SEASON
4G
3G
V:0E
1,1•K/s
VoWiFi STC KSA RIYADH SEASON
Q4. Proof that the probability current for a wavefunction of the form
Þ(x) = R(x) eis(x)
ħ əs(x)
(a) j = |R(x)|²;
m
Əx
ħk
(b) j =
|R|² for a wavevector k.
т
Q5. Proof the following:
(a) Using the Cartesian components of L, Proof that: [Ly . Lz]y = 0
%3|
(b) Lž + L² # (Ly + iL,)(Ly – iL,)
(c) [L². L+] = 0
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