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That's a lie, he never once had hold of the ball.
None whatsoever but there is a pay and display one adjacent.
Port Vale too!
This link was posted by Newbury Blade the other day. It doesnt say if he went to WembleyThat's not fair.
There must be a United fan in his 90's who remembers our win in 1925
My point being that at the macroscopic scale we are used to two broad types of phenomena: waves and particles. Briefly, particles are localised phenomena which transport both mass and energy as they move, while waves are de-localised phenomena (that is they are spread-out in space) which carry energy but no mass as they move. Physical objects that one can touch are particle-like phenomena (e.g. cricket balls), while ripples on a lake (for example) are waves (note that there is no net transport of water: hence no net transport of mass).
If we were to follow the approach suggested by Schrodinger, which was to postulate a function which would vary in both time and space in a wave-like manner (the so-called wavefunction) and which would carry within it information about a particle or system. The time-dependent Schrodinger equation allows us to deterministically predict the behaviour of the wavefunction over time, once we know its environment. The information concerning environment is in the form of the potential which would be experienced by the particle according to classical mechanics.
Whenever we make a measurement on a Quantum system, the results are dictated by the wavefunction at the time at which the measurement is made. It turns out that for each possible quantity we might want to measure (an observable) there is a set of special wavefunctions (known as eigenfunctions) which will always return the same value (an eigenvalue) for the observable. e.g.....
EIGENFUNCTION always returns EIGENVALUE
psi_1(x,t) a_1
psi_2(x,t) a_2
psi_3(x,t) a_3
psi_4(x,t) a_4
etc.... etc....
where (x,t) is standard notation to remind us that the eigenfunctions psi_n(x,t)
are dependent upon position (x) and time (t).
Even if the wavefunction happens not to be one of these eigenfunctions, it is always possible to think of it as a unique superposition of two or more of the eigenfunctions, e.g....
psi(x,t) = c_1*psi_1(x,t) + c_2*psi_2(x,t) + c_3*psi_3(x,t) + ....
where c_1, c_2,.... are coefficients which define the composition of the state.
If a measurement is made on such a state, then the following two things will happen:
The wavefunction will suddenly change into one or other of the eigenfunctions making it up. This is known as the collapse of the wavefunction and the probability of the wavefunction collapsing into a particular eigenfunction depends on how much that eigenfunction contributed to the original superposition. More precisely, the probability that a given eigenfunction will be chosen is proportional to the square of the coefficient of that eigenfunction in the superposition, normalised so that the overall probability of collapse is unity (i.e. the sum of the squares of all the coefficients is 1).
The measurement will return the eigenvalue associated with the eigenfunction into which the wavefunction has collapsed. Clearly therefore the measurement can only ever yield an eigenvalue (even though the original state was not an eigenfunction), and it will do so with a probability determined by the composition of the original superposition. There are clearly only a limited number of discrete values which the observable can take. We say that the system is quantised (which means essentially the same as discretised).
Once the wavefunction has collapsed into one particular eigenfunction it will stay in that state until it is perturbed by the outside world. The fundamental limitation of Quantum Mechanics lies in the Heisenberg Uncertainty Principle which tells us that certain quantum measurements disturb the system and push the wavefunction back into a superposed state once again.
For example, consider a measurement of the position of a particle. Before the measurement is made the particle wavefunction is a superposition of several position eigenfunctions, each corresponding to a different possible position for the particle. When the measurement is made the wavefunction collapses into one of these eigenfunctions, with a probability determined by the composition of the original superposition. One particular position will be recorded by the measurement: the one corresponding to the eigenfunction chosen by the particle.
If a further position measurement is made shortly afterwards the wavefunction will still be the same as when the first measurement was made (because nothing has happened to change it), and so the same position will be recorded. However, if a measurement of the momentum of the particle is now made, the particle wavefunction will change to one of the momentum eigenfunctions (which are not the same as the position eigenfunctions). Thus, if a still later measurement of the position is made, the particle will once again be in a superposition of possible position eigenfunctions, so the position recorded by the measurement will once again come down to probability.
What all this means is that one cannot know both the position and the momentum of a particle at the same time because when you measure one quantity you randomise the value of the other, henceforth while a pound might be too much for tight Yorkshire folk to pay, in affluent areas like Richmond upon Thames and to cash rich folk like your good self, it may be perfectly acceptable. Hope that’s much clearer now.
Applying the same theory to Sundays game, i think we'll win 1-0. Jackson 89' minute free kick
)
Remind me, how do I do long division without a calculator?My point being that at the macroscopic scale we are used to two broad types of phenomena: waves and particles. Briefly, particles are localised phenomena which transport both mass and energy as they move, while waves are de-localised phenomena (that is they are spread-out in space) which carry energy but no mass as they move. Physical objects that one can touch are particle-like phenomena (e.g. cricket balls), while ripples on a lake (for example) are waves (note that there is no net transport of water: hence no net transport of mass).
If we were to follow the approach suggested by Schrodinger, which was to postulate a function which would vary in both time and space in a wave-like manner (the so-called wavefunction) and which would carry within it information about a particle or system. The time-dependent Schrodinger equation allows us to deterministically predict the behaviour of the wavefunction over time, once we know its environment. The information concerning environment is in the form of the potential which would be experienced by the particle according to classical mechanics.
Whenever we make a measurement on a Quantum system, the results are dictated by the wavefunction at the time at which the measurement is made. It turns out that for each possible quantity we might want to measure (an observable) there is a set of special wavefunctions (known as eigenfunctions) which will always return the same value (an eigenvalue) for the observable. e.g.....
EIGENFUNCTION always returns EIGENVALUE
psi_1(x,t) a_1
psi_2(x,t) a_2
psi_3(x,t) a_3
psi_4(x,t) a_4
etc.... etc....
where (x,t) is standard notation to remind us that the eigenfunctions psi_n(x,t)
are dependent upon position (x) and time (t).
Even if the wavefunction happens not to be one of these eigenfunctions, it is always possible to think of it as a unique superposition of two or more of the eigenfunctions, e.g....
psi(x,t) = c_1*psi_1(x,t) + c_2*psi_2(x,t) + c_3*psi_3(x,t) + ....
where c_1, c_2,.... are coefficients which define the composition of the state.
If a measurement is made on such a state, then the following two things will happen:
The wavefunction will suddenly change into one or other of the eigenfunctions making it up. This is known as the collapse of the wavefunction and the probability of the wavefunction collapsing into a particular eigenfunction depends on how much that eigenfunction contributed to the original superposition. More precisely, the probability that a given eigenfunction will be chosen is proportional to the square of the coefficient of that eigenfunction in the superposition, normalised so that the overall probability of collapse is unity (i.e. the sum of the squares of all the coefficients is 1).
The measurement will return the eigenvalue associated with the eigenfunction into which the wavefunction has collapsed. Clearly therefore the measurement can only ever yield an eigenvalue (even though the original state was not an eigenfunction), and it will do so with a probability determined by the composition of the original superposition. There are clearly only a limited number of discrete values which the observable can take. We say that the system is quantised (which means essentially the same as discretised).
Once the wavefunction has collapsed into one particular eigenfunction it will stay in that state until it is perturbed by the outside world. The fundamental limitation of Quantum Mechanics lies in the Heisenberg Uncertainty Principle which tells us that certain quantum measurements disturb the system and push the wavefunction back into a superposed state once again.
For example, consider a measurement of the position of a particle. Before the measurement is made the particle wavefunction is a superposition of several position eigenfunctions, each corresponding to a different possible position for the particle. When the measurement is made the wavefunction collapses into one of these eigenfunctions, with a probability determined by the composition of the original superposition. One particular position will be recorded by the measurement: the one corresponding to the eigenfunction chosen by the particle.
If a further position measurement is made shortly afterwards the wavefunction will still be the same as when the first measurement was made (because nothing has happened to change it), and so the same position will be recorded. However, if a measurement of the momentum of the particle is now made, the particle wavefunction will change to one of the momentum eigenfunctions (which are not the same as the position eigenfunctions). Thus, if a still later measurement of the position is made, the particle will once again be in a superposition of possible position eigenfunctions, so the position recorded by the measurement will once again come down to probability.
What all this means is that one cannot know both the position and the momentum of a particle at the same time because when you measure one quantity you randomise the value of the other, henceforth while a pound might be too much for tight Yorkshire folk to pay, in affluent areas like Richmond upon Thames and to cash rich folk like your good self, it may be perfectly acceptable. Hope that’s much clearer now.
I think quite rightly they may be easing off on some of the segregation.
Still think you might be better parking your car at Meadowhall, right next to Junction 34 of the M1 plently of free parking, get the supertram to Sheffield Station, takes about 10 mins, then 10 mins walk to the ground, then you can leave you son at the station, and get the supertram back to MeadowhallCheers raul and switchblade for the info. Not bothered if its a pay and display, I just need to be parked somewhere near the station so I can put my son on a train after the match, so Eyre lane sounds fine. Thanks again
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