Position Operator
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
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Exercise:
The position operator hat x corresponding to the position x is defined as a multiplication by x. It follows that the expectation value for the position of a quantum mechanical particle e.g. an electron in a state described by the wave function psixt is given by langle x rangle psi^*xt x psixt textrmdx |psixt|^ x textrmdx abcliste abc Show that the expectation value for a particle in a stationary energy eigenstate has the value langle x rangleL/ as expected. abc Derive the uncertay standard deviation for a position measurement if the system is in an energy eigenstate. abcliste
Solution:
abcliste abc The expectation value for the nth energy level is given by langle x rangle A^_^L sin^k_n x x textrmdx fracA^_^Lleft-cosk_n xright x textrmdx fracA^_^L x textrmdx - fracA^ _^Lcos k_n x x textrmdx where we have used the identy sin^alpha fracleft-cosalpharight The first egral can easily be evaluated: _^L x textrmdx fracx^ Big|_^L fracL^ fracL The second egral can be solved through egration by parts with fx x and g'x cosk_n x: _^Lcos k_n x x textrmdx fracsin k_n x k_n x Big|_^L - _^L fracsin k_n x k_n textrmdx fracsin k_n L k_n L + fraccos k_n x k_n^ Big|_^L fracsinpi n k_n L + fraccospi n-cos k_n^ + frac- k_n^ It follows for the expectation value langle x rangle fracA^ fracL^ fracfracLfracL^ fracL abc The uncertay is sigma sqrtlangle x^ rangle - langle x rangle^ We already know that langle x rangle fracL The expectation value for x^ is langle x^ rangle A^_^L sink_n x x^ textrmdx fracA^_^L left-cosk_n xright x^ textrmdx The first term in the egral yields _^L x^ textrmdx fracx^Big|_^L fracL^ For the second term we use egration by parts with fx x^ and g'x cosk_n x: _^L cos k_n x x^ textrmdx fracsin k_n x k_n x^ Big|_^L - _^L fracsin k_n x k_nx textrmdx fracsinpi n-sink_n-_^L fracsin k_n x k_nx textrmdx - _^L fracsin k_n x k_n x textrmdx We use egration by parts again with fx x and g'x sink_n x: dots - leftfrac-cos k_n x k_n^ x Big|_^L - _^L frac-cos k_n x k_n textrmdx right fraccospi nL k_n^+_^L fraccos k_n xk_n textrmdx fracLpi^/L^ + fracsin k_n x k_n^ Big|_^L fracL^ n^ pi^ + fracsinpi-sink_n^ fracL^ n^ pi^ Collecting all the terms we find langle x^ rangle fracA^ leftfracL^ - fracL^ n^ pi^ right fracL L^ leftfrac-frac n^ pi^ right L^ leftfrac-frac n^ pi^right The variance sigma^ is thus sigma^ langle x^ rangle - langle x rangle ^ L^ leftfrac-frac n^ pi^ - frac right L^ leftfrac - frac n^ pi^ right L^ fracn^ pi^ - n^ pi^ and the uncertay sigma sqrtlangle x^ rangle - langle x rangle ^ L sqrtfracn^ pi^ - n^ pi^ For n the relative uncertay is fracsigmaL siP For high quantum numbers ntoinfty the relative uncertay ts to fracsigma_inftyL siinfF siinfP The energy eigenstates are no eigenvectors of the position operator! abcliste
The position operator hat x corresponding to the position x is defined as a multiplication by x. It follows that the expectation value for the position of a quantum mechanical particle e.g. an electron in a state described by the wave function psixt is given by langle x rangle psi^*xt x psixt textrmdx |psixt|^ x textrmdx abcliste abc Show that the expectation value for a particle in a stationary energy eigenstate has the value langle x rangleL/ as expected. abc Derive the uncertay standard deviation for a position measurement if the system is in an energy eigenstate. abcliste
Solution:
abcliste abc The expectation value for the nth energy level is given by langle x rangle A^_^L sin^k_n x x textrmdx fracA^_^Lleft-cosk_n xright x textrmdx fracA^_^L x textrmdx - fracA^ _^Lcos k_n x x textrmdx where we have used the identy sin^alpha fracleft-cosalpharight The first egral can easily be evaluated: _^L x textrmdx fracx^ Big|_^L fracL^ fracL The second egral can be solved through egration by parts with fx x and g'x cosk_n x: _^Lcos k_n x x textrmdx fracsin k_n x k_n x Big|_^L - _^L fracsin k_n x k_n textrmdx fracsin k_n L k_n L + fraccos k_n x k_n^ Big|_^L fracsinpi n k_n L + fraccospi n-cos k_n^ + frac- k_n^ It follows for the expectation value langle x rangle fracA^ fracL^ fracfracLfracL^ fracL abc The uncertay is sigma sqrtlangle x^ rangle - langle x rangle^ We already know that langle x rangle fracL The expectation value for x^ is langle x^ rangle A^_^L sink_n x x^ textrmdx fracA^_^L left-cosk_n xright x^ textrmdx The first term in the egral yields _^L x^ textrmdx fracx^Big|_^L fracL^ For the second term we use egration by parts with fx x^ and g'x cosk_n x: _^L cos k_n x x^ textrmdx fracsin k_n x k_n x^ Big|_^L - _^L fracsin k_n x k_nx textrmdx fracsinpi n-sink_n-_^L fracsin k_n x k_nx textrmdx - _^L fracsin k_n x k_n x textrmdx We use egration by parts again with fx x and g'x sink_n x: dots - leftfrac-cos k_n x k_n^ x Big|_^L - _^L frac-cos k_n x k_n textrmdx right fraccospi nL k_n^+_^L fraccos k_n xk_n textrmdx fracLpi^/L^ + fracsin k_n x k_n^ Big|_^L fracL^ n^ pi^ + fracsinpi-sink_n^ fracL^ n^ pi^ Collecting all the terms we find langle x^ rangle fracA^ leftfracL^ - fracL^ n^ pi^ right fracL L^ leftfrac-frac n^ pi^ right L^ leftfrac-frac n^ pi^right The variance sigma^ is thus sigma^ langle x^ rangle - langle x rangle ^ L^ leftfrac-frac n^ pi^ - frac right L^ leftfrac - frac n^ pi^ right L^ fracn^ pi^ - n^ pi^ and the uncertay sigma sqrtlangle x^ rangle - langle x rangle ^ L sqrtfracn^ pi^ - n^ pi^ For n the relative uncertay is fracsigmaL siP For high quantum numbers ntoinfty the relative uncertay ts to fracsigma_inftyL siinfF siinfP The energy eigenstates are no eigenvectors of the position operator! abcliste
Meta Information
Exercise:
The position operator hat x corresponding to the position x is defined as a multiplication by x. It follows that the expectation value for the position of a quantum mechanical particle e.g. an electron in a state described by the wave function psixt is given by langle x rangle psi^*xt x psixt textrmdx |psixt|^ x textrmdx abcliste abc Show that the expectation value for a particle in a stationary energy eigenstate has the value langle x rangleL/ as expected. abc Derive the uncertay standard deviation for a position measurement if the system is in an energy eigenstate. abcliste
Solution:
abcliste abc The expectation value for the nth energy level is given by langle x rangle A^_^L sin^k_n x x textrmdx fracA^_^Lleft-cosk_n xright x textrmdx fracA^_^L x textrmdx - fracA^ _^Lcos k_n x x textrmdx where we have used the identy sin^alpha fracleft-cosalpharight The first egral can easily be evaluated: _^L x textrmdx fracx^ Big|_^L fracL^ fracL The second egral can be solved through egration by parts with fx x and g'x cosk_n x: _^Lcos k_n x x textrmdx fracsin k_n x k_n x Big|_^L - _^L fracsin k_n x k_n textrmdx fracsin k_n L k_n L + fraccos k_n x k_n^ Big|_^L fracsinpi n k_n L + fraccospi n-cos k_n^ + frac- k_n^ It follows for the expectation value langle x rangle fracA^ fracL^ fracfracLfracL^ fracL abc The uncertay is sigma sqrtlangle x^ rangle - langle x rangle^ We already know that langle x rangle fracL The expectation value for x^ is langle x^ rangle A^_^L sink_n x x^ textrmdx fracA^_^L left-cosk_n xright x^ textrmdx The first term in the egral yields _^L x^ textrmdx fracx^Big|_^L fracL^ For the second term we use egration by parts with fx x^ and g'x cosk_n x: _^L cos k_n x x^ textrmdx fracsin k_n x k_n x^ Big|_^L - _^L fracsin k_n x k_nx textrmdx fracsinpi n-sink_n-_^L fracsin k_n x k_nx textrmdx - _^L fracsin k_n x k_n x textrmdx We use egration by parts again with fx x and g'x sink_n x: dots - leftfrac-cos k_n x k_n^ x Big|_^L - _^L frac-cos k_n x k_n textrmdx right fraccospi nL k_n^+_^L fraccos k_n xk_n textrmdx fracLpi^/L^ + fracsin k_n x k_n^ Big|_^L fracL^ n^ pi^ + fracsinpi-sink_n^ fracL^ n^ pi^ Collecting all the terms we find langle x^ rangle fracA^ leftfracL^ - fracL^ n^ pi^ right fracL L^ leftfrac-frac n^ pi^ right L^ leftfrac-frac n^ pi^right The variance sigma^ is thus sigma^ langle x^ rangle - langle x rangle ^ L^ leftfrac-frac n^ pi^ - frac right L^ leftfrac - frac n^ pi^ right L^ fracn^ pi^ - n^ pi^ and the uncertay sigma sqrtlangle x^ rangle - langle x rangle ^ L sqrtfracn^ pi^ - n^ pi^ For n the relative uncertay is fracsigmaL siP For high quantum numbers ntoinfty the relative uncertay ts to fracsigma_inftyL siinfF siinfP The energy eigenstates are no eigenvectors of the position operator! abcliste
The position operator hat x corresponding to the position x is defined as a multiplication by x. It follows that the expectation value for the position of a quantum mechanical particle e.g. an electron in a state described by the wave function psixt is given by langle x rangle psi^*xt x psixt textrmdx |psixt|^ x textrmdx abcliste abc Show that the expectation value for a particle in a stationary energy eigenstate has the value langle x rangleL/ as expected. abc Derive the uncertay standard deviation for a position measurement if the system is in an energy eigenstate. abcliste
Solution:
abcliste abc The expectation value for the nth energy level is given by langle x rangle A^_^L sin^k_n x x textrmdx fracA^_^Lleft-cosk_n xright x textrmdx fracA^_^L x textrmdx - fracA^ _^Lcos k_n x x textrmdx where we have used the identy sin^alpha fracleft-cosalpharight The first egral can easily be evaluated: _^L x textrmdx fracx^ Big|_^L fracL^ fracL The second egral can be solved through egration by parts with fx x and g'x cosk_n x: _^Lcos k_n x x textrmdx fracsin k_n x k_n x Big|_^L - _^L fracsin k_n x k_n textrmdx fracsin k_n L k_n L + fraccos k_n x k_n^ Big|_^L fracsinpi n k_n L + fraccospi n-cos k_n^ + frac- k_n^ It follows for the expectation value langle x rangle fracA^ fracL^ fracfracLfracL^ fracL abc The uncertay is sigma sqrtlangle x^ rangle - langle x rangle^ We already know that langle x rangle fracL The expectation value for x^ is langle x^ rangle A^_^L sink_n x x^ textrmdx fracA^_^L left-cosk_n xright x^ textrmdx The first term in the egral yields _^L x^ textrmdx fracx^Big|_^L fracL^ For the second term we use egration by parts with fx x^ and g'x cosk_n x: _^L cos k_n x x^ textrmdx fracsin k_n x k_n x^ Big|_^L - _^L fracsin k_n x k_nx textrmdx fracsinpi n-sink_n-_^L fracsin k_n x k_nx textrmdx - _^L fracsin k_n x k_n x textrmdx We use egration by parts again with fx x and g'x sink_n x: dots - leftfrac-cos k_n x k_n^ x Big|_^L - _^L frac-cos k_n x k_n textrmdx right fraccospi nL k_n^+_^L fraccos k_n xk_n textrmdx fracLpi^/L^ + fracsin k_n x k_n^ Big|_^L fracL^ n^ pi^ + fracsinpi-sink_n^ fracL^ n^ pi^ Collecting all the terms we find langle x^ rangle fracA^ leftfracL^ - fracL^ n^ pi^ right fracL L^ leftfrac-frac n^ pi^ right L^ leftfrac-frac n^ pi^right The variance sigma^ is thus sigma^ langle x^ rangle - langle x rangle ^ L^ leftfrac-frac n^ pi^ - frac right L^ leftfrac - frac n^ pi^ right L^ fracn^ pi^ - n^ pi^ and the uncertay sigma sqrtlangle x^ rangle - langle x rangle ^ L sqrtfracn^ pi^ - n^ pi^ For n the relative uncertay is fracsigmaL siP For high quantum numbers ntoinfty the relative uncertay ts to fracsigma_inftyL siinfF siinfP The energy eigenstates are no eigenvectors of the position operator! abcliste
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