Show that psi(-x) is a solution with same eigenvalue as psi(x)
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I have been given the assignment to show that $psi(-x)$ is another solution with the same energy eigenvalue as $psi(x)$, given that the potential is even around $x=0$. The solutions are one-dimensional.
I have tried to get my head around it but I am not sure what to look for. My idea would be to divide through by the wave function in the wave equation, and we would then be left with the energy-eigenvalue $E$ as a function of the potential, which we know is even around the origin:
$$-frachbar^22mfracd^2 psi(x)dx^2 +V(x)psi(x)=psi(x)E$$.
Dividing by $psi(x)$:
$$-frachbar^22mfracd^2dx^2+V(x)=E$$
And the eigenvalue is now a function of the potential only. Is this an illegal move? Or where do I go from here?
quantum-mechanics homework-and-exercises symmetry schroedinger-equation eigenvalue
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I have been given the assignment to show that $psi(-x)$ is another solution with the same energy eigenvalue as $psi(x)$, given that the potential is even around $x=0$. The solutions are one-dimensional.
I have tried to get my head around it but I am not sure what to look for. My idea would be to divide through by the wave function in the wave equation, and we would then be left with the energy-eigenvalue $E$ as a function of the potential, which we know is even around the origin:
$$-frachbar^22mfracd^2 psi(x)dx^2 +V(x)psi(x)=psi(x)E$$.
Dividing by $psi(x)$:
$$-frachbar^22mfracd^2dx^2+V(x)=E$$
And the eigenvalue is now a function of the potential only. Is this an illegal move? Or where do I go from here?
quantum-mechanics homework-and-exercises symmetry schroedinger-equation eigenvalue
New contributor
Jacob Kaldau is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have been given the assignment to show that $psi(-x)$ is another solution with the same energy eigenvalue as $psi(x)$, given that the potential is even around $x=0$. The solutions are one-dimensional.
I have tried to get my head around it but I am not sure what to look for. My idea would be to divide through by the wave function in the wave equation, and we would then be left with the energy-eigenvalue $E$ as a function of the potential, which we know is even around the origin:
$$-frachbar^22mfracd^2 psi(x)dx^2 +V(x)psi(x)=psi(x)E$$.
Dividing by $psi(x)$:
$$-frachbar^22mfracd^2dx^2+V(x)=E$$
And the eigenvalue is now a function of the potential only. Is this an illegal move? Or where do I go from here?
quantum-mechanics homework-and-exercises symmetry schroedinger-equation eigenvalue
New contributor
Jacob Kaldau is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I have been given the assignment to show that $psi(-x)$ is another solution with the same energy eigenvalue as $psi(x)$, given that the potential is even around $x=0$. The solutions are one-dimensional.
I have tried to get my head around it but I am not sure what to look for. My idea would be to divide through by the wave function in the wave equation, and we would then be left with the energy-eigenvalue $E$ as a function of the potential, which we know is even around the origin:
$$-frachbar^22mfracd^2 psi(x)dx^2 +V(x)psi(x)=psi(x)E$$.
Dividing by $psi(x)$:
$$-frachbar^22mfracd^2dx^2+V(x)=E$$
And the eigenvalue is now a function of the potential only. Is this an illegal move? Or where do I go from here?
quantum-mechanics homework-and-exercises symmetry schroedinger-equation eigenvalue
quantum-mechanics homework-and-exercises symmetry schroedinger-equation eigenvalue
New contributor
Jacob Kaldau is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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edited 13 mins ago
Aaron Stevens
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asked 1 hour ago


Jacob Kaldau
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2 Answers
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Simply make a change of variable $xrightarrow -x$ in your equation which becomes
$-frachbar^22mfracd^2dx^2psi(-x)+V(-x)psi(-x)=Epsi(-x)$
Since $V(-x)=V(x)$, this reduces to the same equation for $psi(-x)$ as for $psi(x)$ therefore, they have the same energy eigenvalues.
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First, thank you for taking your time. However, I am completely new to quantum mechanics and I fail to see why the two equation for psi(-x) and psi(x) are the same when the potential is the same. Could you clarify this, or give a hint?
– Jacob Kaldau
41 mins ago
@JacobKaldau This shows that $hat Hpsi(x)=Epsi(x)$ and $hat Hpsi(-x)=Epsi(-x)$, where $hat H$ and $E$ is the same in either case. So $psi(x)$ and $psi(-x)$ have the same eigenvalues $E$ of the same operator $hat H$
– Aaron Stevens
32 mins ago
Ahhh, yes of course. Now I understand it. Thanks a lot! :)
– Jacob Kaldau
24 mins ago
add a comment |Â
up vote
2
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The step you take between your two equations is an illegal operation. This is because dividing $d^2 psi/dx^2$ by $psi(x)$ does not yield $d^2/dx^2$; it yields $psi''(x)/psi(x)$.
An analogous statement would be if you had a scalar $lambda$, a vector $vecv$, and a matrix $A$ such that $A vecv = lambda vecv$. You can't "divide by $vecv$" to yield $A = lambda$.
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Simply make a change of variable $xrightarrow -x$ in your equation which becomes
$-frachbar^22mfracd^2dx^2psi(-x)+V(-x)psi(-x)=Epsi(-x)$
Since $V(-x)=V(x)$, this reduces to the same equation for $psi(-x)$ as for $psi(x)$ therefore, they have the same energy eigenvalues.
New contributor
E. Bellec is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
First, thank you for taking your time. However, I am completely new to quantum mechanics and I fail to see why the two equation for psi(-x) and psi(x) are the same when the potential is the same. Could you clarify this, or give a hint?
– Jacob Kaldau
41 mins ago
@JacobKaldau This shows that $hat Hpsi(x)=Epsi(x)$ and $hat Hpsi(-x)=Epsi(-x)$, where $hat H$ and $E$ is the same in either case. So $psi(x)$ and $psi(-x)$ have the same eigenvalues $E$ of the same operator $hat H$
– Aaron Stevens
32 mins ago
Ahhh, yes of course. Now I understand it. Thanks a lot! :)
– Jacob Kaldau
24 mins ago
add a comment |Â
up vote
3
down vote
Simply make a change of variable $xrightarrow -x$ in your equation which becomes
$-frachbar^22mfracd^2dx^2psi(-x)+V(-x)psi(-x)=Epsi(-x)$
Since $V(-x)=V(x)$, this reduces to the same equation for $psi(-x)$ as for $psi(x)$ therefore, they have the same energy eigenvalues.
New contributor
E. Bellec is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
First, thank you for taking your time. However, I am completely new to quantum mechanics and I fail to see why the two equation for psi(-x) and psi(x) are the same when the potential is the same. Could you clarify this, or give a hint?
– Jacob Kaldau
41 mins ago
@JacobKaldau This shows that $hat Hpsi(x)=Epsi(x)$ and $hat Hpsi(-x)=Epsi(-x)$, where $hat H$ and $E$ is the same in either case. So $psi(x)$ and $psi(-x)$ have the same eigenvalues $E$ of the same operator $hat H$
– Aaron Stevens
32 mins ago
Ahhh, yes of course. Now I understand it. Thanks a lot! :)
– Jacob Kaldau
24 mins ago
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Simply make a change of variable $xrightarrow -x$ in your equation which becomes
$-frachbar^22mfracd^2dx^2psi(-x)+V(-x)psi(-x)=Epsi(-x)$
Since $V(-x)=V(x)$, this reduces to the same equation for $psi(-x)$ as for $psi(x)$ therefore, they have the same energy eigenvalues.
New contributor
E. Bellec is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Simply make a change of variable $xrightarrow -x$ in your equation which becomes
$-frachbar^22mfracd^2dx^2psi(-x)+V(-x)psi(-x)=Epsi(-x)$
Since $V(-x)=V(x)$, this reduces to the same equation for $psi(-x)$ as for $psi(x)$ therefore, they have the same energy eigenvalues.
New contributor
E. Bellec is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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E. Bellec is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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answered 50 mins ago
E. Bellec
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311
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First, thank you for taking your time. However, I am completely new to quantum mechanics and I fail to see why the two equation for psi(-x) and psi(x) are the same when the potential is the same. Could you clarify this, or give a hint?
– Jacob Kaldau
41 mins ago
@JacobKaldau This shows that $hat Hpsi(x)=Epsi(x)$ and $hat Hpsi(-x)=Epsi(-x)$, where $hat H$ and $E$ is the same in either case. So $psi(x)$ and $psi(-x)$ have the same eigenvalues $E$ of the same operator $hat H$
– Aaron Stevens
32 mins ago
Ahhh, yes of course. Now I understand it. Thanks a lot! :)
– Jacob Kaldau
24 mins ago
add a comment |Â
First, thank you for taking your time. However, I am completely new to quantum mechanics and I fail to see why the two equation for psi(-x) and psi(x) are the same when the potential is the same. Could you clarify this, or give a hint?
– Jacob Kaldau
41 mins ago
@JacobKaldau This shows that $hat Hpsi(x)=Epsi(x)$ and $hat Hpsi(-x)=Epsi(-x)$, where $hat H$ and $E$ is the same in either case. So $psi(x)$ and $psi(-x)$ have the same eigenvalues $E$ of the same operator $hat H$
– Aaron Stevens
32 mins ago
Ahhh, yes of course. Now I understand it. Thanks a lot! :)
– Jacob Kaldau
24 mins ago
First, thank you for taking your time. However, I am completely new to quantum mechanics and I fail to see why the two equation for psi(-x) and psi(x) are the same when the potential is the same. Could you clarify this, or give a hint?
– Jacob Kaldau
41 mins ago
First, thank you for taking your time. However, I am completely new to quantum mechanics and I fail to see why the two equation for psi(-x) and psi(x) are the same when the potential is the same. Could you clarify this, or give a hint?
– Jacob Kaldau
41 mins ago
@JacobKaldau This shows that $hat Hpsi(x)=Epsi(x)$ and $hat Hpsi(-x)=Epsi(-x)$, where $hat H$ and $E$ is the same in either case. So $psi(x)$ and $psi(-x)$ have the same eigenvalues $E$ of the same operator $hat H$
– Aaron Stevens
32 mins ago
@JacobKaldau This shows that $hat Hpsi(x)=Epsi(x)$ and $hat Hpsi(-x)=Epsi(-x)$, where $hat H$ and $E$ is the same in either case. So $psi(x)$ and $psi(-x)$ have the same eigenvalues $E$ of the same operator $hat H$
– Aaron Stevens
32 mins ago
Ahhh, yes of course. Now I understand it. Thanks a lot! :)
– Jacob Kaldau
24 mins ago
Ahhh, yes of course. Now I understand it. Thanks a lot! :)
– Jacob Kaldau
24 mins ago
add a comment |Â
up vote
2
down vote
The step you take between your two equations is an illegal operation. This is because dividing $d^2 psi/dx^2$ by $psi(x)$ does not yield $d^2/dx^2$; it yields $psi''(x)/psi(x)$.
An analogous statement would be if you had a scalar $lambda$, a vector $vecv$, and a matrix $A$ such that $A vecv = lambda vecv$. You can't "divide by $vecv$" to yield $A = lambda$.
add a comment |Â
up vote
2
down vote
The step you take between your two equations is an illegal operation. This is because dividing $d^2 psi/dx^2$ by $psi(x)$ does not yield $d^2/dx^2$; it yields $psi''(x)/psi(x)$.
An analogous statement would be if you had a scalar $lambda$, a vector $vecv$, and a matrix $A$ such that $A vecv = lambda vecv$. You can't "divide by $vecv$" to yield $A = lambda$.
add a comment |Â
up vote
2
down vote
up vote
2
down vote
The step you take between your two equations is an illegal operation. This is because dividing $d^2 psi/dx^2$ by $psi(x)$ does not yield $d^2/dx^2$; it yields $psi''(x)/psi(x)$.
An analogous statement would be if you had a scalar $lambda$, a vector $vecv$, and a matrix $A$ such that $A vecv = lambda vecv$. You can't "divide by $vecv$" to yield $A = lambda$.
The step you take between your two equations is an illegal operation. This is because dividing $d^2 psi/dx^2$ by $psi(x)$ does not yield $d^2/dx^2$; it yields $psi''(x)/psi(x)$.
An analogous statement would be if you had a scalar $lambda$, a vector $vecv$, and a matrix $A$ such that $A vecv = lambda vecv$. You can't "divide by $vecv$" to yield $A = lambda$.
answered 46 mins ago


Michael Seifert
13.7k12651
13.7k12651
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