The odd parity eigenstates of the infinte square well , with potential V = 0 in the range −L/2 ≤ x ≤ L/2, are given by : (see figure) and have Ψn(x, t) = 0 elsewhere , for n=2 , 4 , 6 etc I have got the expectation value of momentum for ⟨p⟩ and ⟨p 2⟩ for n = 2 (see figures) Determine the uncertainty in momentum, ∆p, for a particle with n = 2, and use your result to put a lower bound on the uncertainty in position via Heisenberg’s uncertainty relation.
The odd parity eigenstates of the infinte square well , with potential V = 0 in the range −L/2 ≤ x ≤ L/2, are given by : (see figure) and have Ψn(x, t) = 0 elsewhere , for n=2 , 4 , 6 etc I have got the expectation value of momentum for ⟨p⟩ and ⟨p 2⟩ for n = 2 (see figures) Determine the uncertainty in momentum, ∆p, for a particle with n = 2, and use your result to put a lower bound on the uncertainty in position via Heisenberg’s uncertainty relation.
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The odd parity eigenstates of the infinte square well , with potential V = 0 in the range −L/2 ≤ x ≤ L/2, are given by :
(see figure)
and have Ψn(x, t) = 0 elsewhere , for n=2 , 4 , 6 etc
I have got the expectation value of momentum for ⟨p⟩ and ⟨p 2⟩ for n = 2 (see figures)
Determine the uncertainty in momentum, ∆p, for a particle with n = 2, and use your result to put a lower bound on the uncertainty in position via Heisenberg’s uncertainty relation.
Transcribed Image Text:√sin (2)
and have ₂(x, t) = 0 elsewhere, for n = 2, 4, 6, etc.
Yn(x, t)
=
-iEnt/ħ
for - sest
![p=770x
and
+(x, t) = √sin (17x)e-Math
‘xle-iEnt/
L
(nπ
**(x1) = √sin(x)+1;//A
t
e+iEnt/
So, the expectation value of momentum is defined as,
p> = √+ x 4₁₂(x, t). p.4₁" (x, t)dx
<
+ sin(x)/x sin(x) dx
h
e+iEnt/
L
2-i
cos (17x). · e-iEnt/ h +iEnt/ h dx
=
II
=
=
-2i nπ
Lħ L
(c) Momentum operator is,
-i a
<p> = ħ ax
So,
p² =
and
nπ
(x, 1) = √√sin (1x)e-186,0/ h
**(x, 1) = √sin (1x) + h
=
||
II
So, the expectation value of momentum is defined as,
p² > = ƒ± % 4₁(x, t). p² · 4₁ *(x, t)dx
√sin(x)=√sin(x)e+://dx
-1
e-Ent/ h
=
+L/2,
h-L₂Sin
-18²
ħ20x²
=
.0
+ ∞o
-2i nπ+L/2
Lh
7+1/²sin (17x) · cos (17x) . 1. dx
nπ
2
Lh
nπ
2
2/2+1/2 - sin (17x). (17) ²
-L/2
2
(sin(x) sin(x) - 1. dx
2
Lh
nπ
nπ
2² (1) S+L/2 in ¹ (1x) dx
Lh
nπ
Lh
22 - (1277)² - 1/1
Lh
. sin
2
nπ
²² (1) ²/1/21 - 008 (2127x)].
(2nπ
- dx
Lh
L
11/1/1
nπ
[0-71 (2)
( odd function)
-
(17x). · e-iEnt/ h + iEnt/ ħ dx
.
x-sin
-L/2
(2nπ
2nπ
2²2 · (~7-)² · |- - sin (²nr. 4) - [ - - - sin (²2nr - (-:-))]]
Lħ²
L
(2nπ
L
2nπ
(7) sin(²77)+ -sin (²)]
(2nπ
2
L
L
h
+L/2
nπ
2nπ
12 (17)²2 / L - 2sin (²172)
L
Lh
(17)[L-2sin (nn)]
Lh
(n = 1, 2, 3, ...)
(NT) ²
It is also true for every value of 'n' (i.e. n=1, 2, 3, 4)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b765a29-fcb0-4895-9cef-5ef60c256f5b%2F95430d75-e9f3-4a34-b3be-e7e476439bf9%2Feedz2e5i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:p=770x
and
+(x, t) = √sin (17x)e-Math
‘xle-iEnt/
L
(nπ
**(x1) = √sin(x)+1;//A
t
e+iEnt/
So, the expectation value of momentum is defined as,
p> = √+ x 4₁₂(x, t). p.4₁" (x, t)dx
<
+ sin(x)/x sin(x) dx
h
e+iEnt/
L
2-i
cos (17x). · e-iEnt/ h +iEnt/ h dx
=
II
=
=
-2i nπ
Lħ L
(c) Momentum operator is,
-i a
<p> = ħ ax
So,
p² =
and
nπ
(x, 1) = √√sin (1x)e-186,0/ h
**(x, 1) = √sin (1x) + h
=
||
II
So, the expectation value of momentum is defined as,
p² > = ƒ± % 4₁(x, t). p² · 4₁ *(x, t)dx
√sin(x)=√sin(x)e+://dx
-1
e-Ent/ h
=
+L/2,
h-L₂Sin
-18²
ħ20x²
=
.0
+ ∞o
-2i nπ+L/2
Lh
7+1/²sin (17x) · cos (17x) . 1. dx
nπ
2
Lh
nπ
2
2/2+1/2 - sin (17x). (17) ²
-L/2
2
(sin(x) sin(x) - 1. dx
2
Lh
nπ
nπ
2² (1) S+L/2 in ¹ (1x) dx
Lh
nπ
Lh
22 - (1277)² - 1/1
Lh
. sin
2
nπ
²² (1) ²/1/21 - 008 (2127x)].
(2nπ
- dx
Lh
L
11/1/1
nπ
[0-71 (2)
( odd function)
-
(17x). · e-iEnt/ h + iEnt/ ħ dx
.
x-sin
-L/2
(2nπ
2nπ
2²2 · (~7-)² · |- - sin (²nr. 4) - [ - - - sin (²2nr - (-:-))]]
Lħ²
L
(2nπ
L
2nπ
(7) sin(²77)+ -sin (²)]
(2nπ
2
L
L
h
+L/2
nπ
2nπ
12 (17)²2 / L - 2sin (²172)
L
Lh
(17)[L-2sin (nn)]
Lh
(n = 1, 2, 3, ...)
(NT) ²
It is also true for every value of 'n' (i.e. n=1, 2, 3, 4)
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