Ground State Statistics
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
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
The wave function for the ground state of a hydrogen atom is given by psi_r frace^-r/a_pi a_^/ abcliste abc Show that psi_ is normalised i.e. that the volume egral of its square modulus is equal to : _V |psi_r|^ r^ textrmdV abc The radial probability density for a radially symmetrical wave function is Pr pi r^ |psi_r|^ Calculate the radius with the highest probability density. abc Calculate the expectation value for the radius operator hat r which is defined through its action on the wave function: hat r psi_r r psi_r abc Calculate the uncertay for the radius operator. abcliste
Solution:
abcliste abc _V |psi_r| r^ textrmdV pi^_^infty |frace^-r/a_pi a_^/|^ r^ textrmdr fracpi^pi^ a_^ _^infty r^ lefte^-r/a_right textrmdr fraca_^frac a_^ quad square where we have used the general expression for the egral I'_n _^infty r^n lefte^-r/aright^ textrmdr fracn! a^n+^n+ see exercises for n. abc The radial probability density for the ground state is Pr pi r^ |psi_r|^ fracpi r^pi^ a_^ e^-r/a_ fraca_^ r^ e^-r/a_ This expression has extremal pos for P'r Longrightarrow fractextrmdtextrmdrleftr^ e^-r/a_right r e^-r/a_ - r^ e^-r/a_-/a_ e^-r/a_leftr-fracr^a_right Longrightarrow r r-a_ Longrightarrow r quad textrmor quad r a_ The solution r is a minimum psi_ but ra_ is the maximum. It follows that the Bohr radius a_ is the radius with the highest radial probability in the ground state. abc The expectation value for the radius operator is langle hat r rangle _V psi^*_r r psi_r r^ textrmdV pi^ _^infty |psi_r|^ r r^ textrmdr fracpi^pi^ a_^ _^infty e^-r/a_ r^ textrmdr fraca_^ frac a_^^ fraca_ The expectation value is / times the Bohr radius. abc The expectation value for the square of the radius operator can be calculated in the same way: langle hat r^ rangle _V psi^*_r r^ psi_r r^ textrmdV pi^ _^infty |psi_r|^ r^ r^ textrmdr fracpi^pi^ a_^ _^infty e^-r/a_ r^ textrmdr fraca_^ frac a_^ a_^ It follows for the uncertay standard deviation of the radius sigma sqrtlangle hat r^ rangllangle hat r rangle^ sqrt a_^-leftfrac a_right^ sqrtfraca_^ fracsqrt a_ abcliste The calculations can be verified or exted to other states using the Mathematica file linked to from this exercise.
The wave function for the ground state of a hydrogen atom is given by psi_r frace^-r/a_pi a_^/ abcliste abc Show that psi_ is normalised i.e. that the volume egral of its square modulus is equal to : _V |psi_r|^ r^ textrmdV abc The radial probability density for a radially symmetrical wave function is Pr pi r^ |psi_r|^ Calculate the radius with the highest probability density. abc Calculate the expectation value for the radius operator hat r which is defined through its action on the wave function: hat r psi_r r psi_r abc Calculate the uncertay for the radius operator. abcliste
Solution:
abcliste abc _V |psi_r| r^ textrmdV pi^_^infty |frace^-r/a_pi a_^/|^ r^ textrmdr fracpi^pi^ a_^ _^infty r^ lefte^-r/a_right textrmdr fraca_^frac a_^ quad square where we have used the general expression for the egral I'_n _^infty r^n lefte^-r/aright^ textrmdr fracn! a^n+^n+ see exercises for n. abc The radial probability density for the ground state is Pr pi r^ |psi_r|^ fracpi r^pi^ a_^ e^-r/a_ fraca_^ r^ e^-r/a_ This expression has extremal pos for P'r Longrightarrow fractextrmdtextrmdrleftr^ e^-r/a_right r e^-r/a_ - r^ e^-r/a_-/a_ e^-r/a_leftr-fracr^a_right Longrightarrow r r-a_ Longrightarrow r quad textrmor quad r a_ The solution r is a minimum psi_ but ra_ is the maximum. It follows that the Bohr radius a_ is the radius with the highest radial probability in the ground state. abc The expectation value for the radius operator is langle hat r rangle _V psi^*_r r psi_r r^ textrmdV pi^ _^infty |psi_r|^ r r^ textrmdr fracpi^pi^ a_^ _^infty e^-r/a_ r^ textrmdr fraca_^ frac a_^^ fraca_ The expectation value is / times the Bohr radius. abc The expectation value for the square of the radius operator can be calculated in the same way: langle hat r^ rangle _V psi^*_r r^ psi_r r^ textrmdV pi^ _^infty |psi_r|^ r^ r^ textrmdr fracpi^pi^ a_^ _^infty e^-r/a_ r^ textrmdr fraca_^ frac a_^ a_^ It follows for the uncertay standard deviation of the radius sigma sqrtlangle hat r^ rangllangle hat r rangle^ sqrt a_^-leftfrac a_right^ sqrtfraca_^ fracsqrt a_ abcliste The calculations can be verified or exted to other states using the Mathematica file linked to from this exercise.
Meta Information
Exercise:
The wave function for the ground state of a hydrogen atom is given by psi_r frace^-r/a_pi a_^/ abcliste abc Show that psi_ is normalised i.e. that the volume egral of its square modulus is equal to : _V |psi_r|^ r^ textrmdV abc The radial probability density for a radially symmetrical wave function is Pr pi r^ |psi_r|^ Calculate the radius with the highest probability density. abc Calculate the expectation value for the radius operator hat r which is defined through its action on the wave function: hat r psi_r r psi_r abc Calculate the uncertay for the radius operator. abcliste
Solution:
abcliste abc _V |psi_r| r^ textrmdV pi^_^infty |frace^-r/a_pi a_^/|^ r^ textrmdr fracpi^pi^ a_^ _^infty r^ lefte^-r/a_right textrmdr fraca_^frac a_^ quad square where we have used the general expression for the egral I'_n _^infty r^n lefte^-r/aright^ textrmdr fracn! a^n+^n+ see exercises for n. abc The radial probability density for the ground state is Pr pi r^ |psi_r|^ fracpi r^pi^ a_^ e^-r/a_ fraca_^ r^ e^-r/a_ This expression has extremal pos for P'r Longrightarrow fractextrmdtextrmdrleftr^ e^-r/a_right r e^-r/a_ - r^ e^-r/a_-/a_ e^-r/a_leftr-fracr^a_right Longrightarrow r r-a_ Longrightarrow r quad textrmor quad r a_ The solution r is a minimum psi_ but ra_ is the maximum. It follows that the Bohr radius a_ is the radius with the highest radial probability in the ground state. abc The expectation value for the radius operator is langle hat r rangle _V psi^*_r r psi_r r^ textrmdV pi^ _^infty |psi_r|^ r r^ textrmdr fracpi^pi^ a_^ _^infty e^-r/a_ r^ textrmdr fraca_^ frac a_^^ fraca_ The expectation value is / times the Bohr radius. abc The expectation value for the square of the radius operator can be calculated in the same way: langle hat r^ rangle _V psi^*_r r^ psi_r r^ textrmdV pi^ _^infty |psi_r|^ r^ r^ textrmdr fracpi^pi^ a_^ _^infty e^-r/a_ r^ textrmdr fraca_^ frac a_^ a_^ It follows for the uncertay standard deviation of the radius sigma sqrtlangle hat r^ rangllangle hat r rangle^ sqrt a_^-leftfrac a_right^ sqrtfraca_^ fracsqrt a_ abcliste The calculations can be verified or exted to other states using the Mathematica file linked to from this exercise.
The wave function for the ground state of a hydrogen atom is given by psi_r frace^-r/a_pi a_^/ abcliste abc Show that psi_ is normalised i.e. that the volume egral of its square modulus is equal to : _V |psi_r|^ r^ textrmdV abc The radial probability density for a radially symmetrical wave function is Pr pi r^ |psi_r|^ Calculate the radius with the highest probability density. abc Calculate the expectation value for the radius operator hat r which is defined through its action on the wave function: hat r psi_r r psi_r abc Calculate the uncertay for the radius operator. abcliste
Solution:
abcliste abc _V |psi_r| r^ textrmdV pi^_^infty |frace^-r/a_pi a_^/|^ r^ textrmdr fracpi^pi^ a_^ _^infty r^ lefte^-r/a_right textrmdr fraca_^frac a_^ quad square where we have used the general expression for the egral I'_n _^infty r^n lefte^-r/aright^ textrmdr fracn! a^n+^n+ see exercises for n. abc The radial probability density for the ground state is Pr pi r^ |psi_r|^ fracpi r^pi^ a_^ e^-r/a_ fraca_^ r^ e^-r/a_ This expression has extremal pos for P'r Longrightarrow fractextrmdtextrmdrleftr^ e^-r/a_right r e^-r/a_ - r^ e^-r/a_-/a_ e^-r/a_leftr-fracr^a_right Longrightarrow r r-a_ Longrightarrow r quad textrmor quad r a_ The solution r is a minimum psi_ but ra_ is the maximum. It follows that the Bohr radius a_ is the radius with the highest radial probability in the ground state. abc The expectation value for the radius operator is langle hat r rangle _V psi^*_r r psi_r r^ textrmdV pi^ _^infty |psi_r|^ r r^ textrmdr fracpi^pi^ a_^ _^infty e^-r/a_ r^ textrmdr fraca_^ frac a_^^ fraca_ The expectation value is / times the Bohr radius. abc The expectation value for the square of the radius operator can be calculated in the same way: langle hat r^ rangle _V psi^*_r r^ psi_r r^ textrmdV pi^ _^infty |psi_r|^ r^ r^ textrmdr fracpi^pi^ a_^ _^infty e^-r/a_ r^ textrmdr fraca_^ frac a_^ a_^ It follows for the uncertay standard deviation of the radius sigma sqrtlangle hat r^ rangllangle hat r rangle^ sqrt a_^-leftfrac a_right^ sqrtfraca_^ fracsqrt a_ abcliste The calculations can be verified or exted to other states using the Mathematica file linked to from this exercise.
Contained in these collections:
-
Hydrogen Atom by by