- #1
Lasse Jepsen
- 2
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Moved from a technical forum, so homework template missing
The question is as follows:
A particle of mass m has the wave function
psi(x, t) = A * e^( -a ( ( m*x^2 / hbar) +i*t ) )
where A and a are positive real constants.
i don't know how to format my stuff on this website, so it may be a bit harder to read. Generally when i write "int" i mean the integral from -infinity to +infinity.
first it asks me to find A, i do this by normalizing the wave function, and i find that
A = (2*a*m / (pi * hbar) )^( 1 / 4)
this is correct according to the solutions manual.
i'm then asked to find the potential energy function, but since this doesn't mean anything for the rest of the question i will ignore it.
i am then asked to calculate the expectation values of x, x^2, p and p^2
so starting with the expectation value of x (i will write this as <x>) :
from calculating A i already know that int( e^( -2*a ( (m*x^2 / hbar) ) dx) = ( pi * hbar / (2*a*m) )^( 1 / 2) = A^( -2 )
<x> = int( x * |psi(x, t)|^2 dx) = A^2 int( x * e^( -2*a ( (m*x^2 / hbar) ) dx)
now i use integration by parts
int( u dv) = u*v - int( v du)
i use the following u and v:
u = x
du = dx
dv = e^( -2*a ( (m*x^2 / hbar) ) dx
v = int( dv ) = int( e^( -2*a ( (m*x^2 / hbar) ) dx) = A^( -2 )
so my integral becomes:
int( x * e^( -2*a ( (m*x^2 / hbar) ) dx) = x * A^( -2 ) - int( A^( -2 ) dx) = x * A^( -2 ) - A^( -2 ) * x = 0 = <x>
this means <x> = A^2 * 0 = 0
the solutions manual agrees with me on this solution.
but calculating <x^2> doesn't give the the correct answer, and this makes me question whether i use integration by parts wrong.
i'll show you my result for <x^2>
<x^2> = int( x^2 * |psi(x, t)|^2 dx) = A^2 int( x^2 * e^( -2*a ( (m*x^2 / hbar) ) dx)
now i use integration by parts
int( u dv) = u*v - int( v du)
i use the following u and v:
u = x^2
du = 2 * x dx
dv = e^( -2*a ( (m*x^2 / hbar) ) dx
v = int( dv ) = int( e^( -2*a ( (m*x^2 / hbar) ) dx) = A^( -2 )
so my integral becomes:
int( x^2 * e^( -2*a ( (m*x^2 / hbar) ) dx) = x^2 * A^( -2 ) - int( A^( -2 ) * 2 * x dx) = x^2 * A^( -2 ) - A^( -2 ) * 2 * int( x dx)
= x^2 * A^( -2 ) - A^( -2 ) * 2 * 1/2 * x^2 = 0 = <x^2>
but according to the solutions manual i should get
<x^2> = hbar / (4*a*m)
no matter what i do i can't seem to figure out what I'm doing wrong please help.
for anyone interested this is problem 1.9 from David J. Griffiths "introduction to Quantum Mechanics Third edition"
A particle of mass m has the wave function
psi(x, t) = A * e^( -a ( ( m*x^2 / hbar) +i*t ) )
where A and a are positive real constants.
i don't know how to format my stuff on this website, so it may be a bit harder to read. Generally when i write "int" i mean the integral from -infinity to +infinity.
first it asks me to find A, i do this by normalizing the wave function, and i find that
A = (2*a*m / (pi * hbar) )^( 1 / 4)
this is correct according to the solutions manual.
i'm then asked to find the potential energy function, but since this doesn't mean anything for the rest of the question i will ignore it.
i am then asked to calculate the expectation values of x, x^2, p and p^2
so starting with the expectation value of x (i will write this as <x>) :
from calculating A i already know that int( e^( -2*a ( (m*x^2 / hbar) ) dx) = ( pi * hbar / (2*a*m) )^( 1 / 2) = A^( -2 )
<x> = int( x * |psi(x, t)|^2 dx) = A^2 int( x * e^( -2*a ( (m*x^2 / hbar) ) dx)
now i use integration by parts
int( u dv) = u*v - int( v du)
i use the following u and v:
u = x
du = dx
dv = e^( -2*a ( (m*x^2 / hbar) ) dx
v = int( dv ) = int( e^( -2*a ( (m*x^2 / hbar) ) dx) = A^( -2 )
so my integral becomes:
int( x * e^( -2*a ( (m*x^2 / hbar) ) dx) = x * A^( -2 ) - int( A^( -2 ) dx) = x * A^( -2 ) - A^( -2 ) * x = 0 = <x>
this means <x> = A^2 * 0 = 0
the solutions manual agrees with me on this solution.
but calculating <x^2> doesn't give the the correct answer, and this makes me question whether i use integration by parts wrong.
i'll show you my result for <x^2>
<x^2> = int( x^2 * |psi(x, t)|^2 dx) = A^2 int( x^2 * e^( -2*a ( (m*x^2 / hbar) ) dx)
now i use integration by parts
int( u dv) = u*v - int( v du)
i use the following u and v:
u = x^2
du = 2 * x dx
dv = e^( -2*a ( (m*x^2 / hbar) ) dx
v = int( dv ) = int( e^( -2*a ( (m*x^2 / hbar) ) dx) = A^( -2 )
so my integral becomes:
int( x^2 * e^( -2*a ( (m*x^2 / hbar) ) dx) = x^2 * A^( -2 ) - int( A^( -2 ) * 2 * x dx) = x^2 * A^( -2 ) - A^( -2 ) * 2 * int( x dx)
= x^2 * A^( -2 ) - A^( -2 ) * 2 * 1/2 * x^2 = 0 = <x^2>
but according to the solutions manual i should get
<x^2> = hbar / (4*a*m)
no matter what i do i can't seem to figure out what I'm doing wrong please help.
for anyone interested this is problem 1.9 from David J. Griffiths "introduction to Quantum Mechanics Third edition"