Additionally, in quantum mechanics those observable quatities can be discrete or continuous. My answer hopefully provides, by a progressive series of simple statements and examples, a way of actually conceiving what a wave function is. You may want to scroll to the final paragraph. A function is a mathematical representation of something. A wave function is a mathematical representation of a wave. I'm sure I will be dissed for answering your question literally. I do have 4 years of tertiary maths and 3 years of tertiary physics behind me, but you did seem to want a very simple example.
An electron is sometimes represented as somewhat like a short burst of waves, and the complex formula for this "shape" is called the wave function. Your quote from the textbook is not a definition of a wave function, it is telling you something about the wave function representing an electron.
You can imagine a straight line with a burst of wave activity somewhere along its length. If the line represents distance, then the position of the burst is the location of the electron. If the line represents time then it indicates when the electron was at some location. The wave function of an electron describes the time, location, and in fact everything we could know about the electron. Because any method of looking at the electron has an effect on it rather like using a torch to find out what rabbits do in the dark , we never really know exactly every value of every parameter associated with the electron.
The wave equation therefore can not simply tell us where the electron is, or how fast it is going. It gives us the probabilities of the electron or other phenomenon being at any location at any speed. In our everyday experience, the way we imagine particles to be is quite adequate. It is easy to forget that this conception of a particle is a model, not a reality.
In fact there is no such thing as a distinct surface of a particle. When we are dealing with things at a sub-atomic level, our everyday concept breaks down, and we admit that we only know probabilities of the where-and-when of a particle. The wave function for an electron therefore includes everything we know about the electron, which includes that it is not a particle in the everyday sense, but is an array of greater and lesser probabilities of being somewhere at some time.
The complexity is not a reflection of reality, but rather of how difficult it is to create a model than can be understood by a mind that generally needs to relate to what can be perceived by the senses.
Sign up to join this community. In general, a qubit is not in a state of zero or one, but rather in a mixed state of zero and one. If a large number of qubits are placed in the same quantum state, the measurement of an individual qubit would produce a zero with a probability p , and a one with a probability Many scientists believe that quantum computers are the future of the computer industry.
The specific form of the wave function depends on the details of the physical system. A peculiarity of quantum theory is that these functions are usually complex function s. A complex function is one that contains one or more imaginary numbers. Experimental measurements produce real nonimaginary numbers only, so the above procedure to use the wave function must be slightly modified. The complex conjugate of a function is obtaining by replacing every occurrence of in that function with.
This procedure eliminates complex numbers in all predictions because the product is always a real number. Check Your Understanding If , what is the product? Consider the motion of a free particle that moves along the x -direction. As the name suggests, a free particle experiences no forces and so moves with a constant velocity. As we will see in a later section of this chapter, a formal quantum mechanical treatment of a free particle indicates that its wave function has real and complex parts.
In particular, the wave function is given by. If the wave function varies slowly over the interval the probability of finding the particle in that interval is.
If A has real and complex parts , where a and b are real constants , then. Check Your Understanding Suppose that a particle with energy E is moving along the x -axis and is confined in the region between 0 and L.
One possible wave function is. In classical mechanics, the solution to an equation of motion is a function of a measurable quantity, such as x t , where x is the position and t is the time. Note that the particle has one value of position for any time t.
In quantum mechanics, however, the solution to an equation of motion is a wave function, The particle has many values of position for any time t , and only the probability density of finding the particle, , can be known. The average value of position for a large number of particles with the same wave function is expected to be. This is called the expectation value of the position. It is usually written. The reason for this will become apparent soon.
Formally, x is called the position operator. At this point, it is important to stress that a wave function can be written in terms of other quantities as well, such as velocity v , momentum p , and kinetic energy K. The expectation value of momentum, for example, can be written. Where dp is used instead of dx to indicate an infinitesimal interval in momentum.
In some cases, we know the wave function in position, but seek the expectation of momentum. The procedure for doing this is. The momentum operator in the x -direction is sometimes denoted.
Momentum operators for the y — and z -directions are defined similarly. This operator and many others are derived in a more advanced course in modern physics. In some cases, this derivation is relatively simple. For example, the kinetic energy operator is just. Thus, if we seek an expectation value of kinetic energy of a particle in one dimension, two successive ordinary derivatives of the wave function are required before integration.
Expectation-value calculations are often simplified by exploiting the symmetry of wave functions. Symmetric wave functions can be even or odd.
An even function is a function that satisfies. In contrast, an odd function is a function that satisfies. An example of even and odd functions is shown in Figure. An even function is symmetric about the y -axis. By comparison, an odd function is generated by reflecting the function about the y -axis and then about the x -axis.
An odd function is also referred to as an anti-symmetric function. In general, an even function times an even function produces an even function. A simple example of an even function is the product even times even is even.
Similarly, an odd function times an odd function produces an even function, such as x sin x odd times odd is even. Soon, you will learn soon that the wavefunction can be used to make many other kinds of predictions, as well.
This answer is called the Copenhagen interpretation of the wavefunction, or of quantum mechanics. The assumption that a particle can only have one value of position when the observer is not looking is abandoned.
Similar comments can be made of other measurable quantities, such as momentum and energy. When the radioactive substance decays, the Geiger detects it and triggers the hammer to release the poison, which subsequently kills the cat. The radioactive decay is a random [probabilistic] process, and there is no way to predict when it will happen.
Physicists say the atom exists in a state known as a superposition—both decayed and not decayed at the same time.
However, this interpretation remains the most commonly taught view of quantum mechanics. Two-state systems left and right, atom decays and does not decay, and so on are often used to illustrate the principles of quantum mechanics. These systems find many applications in nature, including electron spin and mixed states of particles, atoms, and even molecules. Two-state systems are also finding application in the quantum computer, as mentioned in the introduction of this chapter.
Unlike a digital computer, which encodes information in binary digits zeroes and ones , a quantum computer stores and manipulates data in the form of quantum bits, or qubits. In general, a qubit is not in a state of zero or one, but rather in a mixed state of zero and one.
Some scientists believe that quantum computers are the future of the computer industry. The specific form of the wavefunction depends on the details of the physical system. A peculiarity of quantum theory is that these functions are usually complex functions.
Experimental measurements produce real nonimaginary numbers only, so the above procedure to use the wavefunction must be slightly modified. Consider the motion of a free particle that moves along the x -direction. As the name suggests, a free particle experiences no forces and so moves with a constant velocity. As we will see in a later section of this chapter, a formal quantum mechanical treatment of a free particle indicates that its wavefunction has real and complex parts.
In particular, the wavefunction is given by. Suppose that a particle with energy E is moving along the x -axis and is confined in the region between 0 and L. One possible wavefunction is. The average value of position for a large number of particles with the same wavefunction is expected to be. This is called the expectation value of the position. It is usually written. The reason for this will become apparent soon. The expectation value of momentum, for example, can be written.
In , Niels Bohr and others advocated this alternative view in the Copenhagen interpretation, in which the wave function is merely a mathematical probability that immediately assumes only one value when an observer measures the system, resulting in the wave function collapsing. Still others disagree with both views: in the '30s, Einstein, Podolsky, and Rosen argued that the wave function does not provide a complete physical description of reality and suggested that the entire theory of quantum mechanics is incomplete.
In their paper, Colbeck and Renner illustrate the difference between the two main views of the wave function's probabilistic nature with a simple example:.
The fact that the prediction is probabilistic then solely reflects a lack of knowledge on the part of the meteorologist on these conditions.
In particular, the forecast is not an element of reality associated with the atmosphere but rather reflects the subjective knowledge of the forecaster; a second meteorologist with different knowledge may issue an alternative forecast.
Moving to quantum mechanics, one may ask whether the wave function that we assign to a quantum system should be seen as a subjective object analogous to the weather forecast representing the knowledge an experimenter has about the system or whether the wave function is an element of reality of the system analogous to the weather being sunny.
Colbeck and Renner argue that, unlike a weather forecast, the wave function of a quantum system fully describes reality itself, not simply a physicist's lack of knowledge of reality. The claim's only assumptions are that measurement settings can be freely chosen and that quantum theory gives the correct statistical predictions, both of which are usually implicit in physics research, as well as experimentally falsifiable. This means that there is inherent randomness in nature. The scientists' claim relies on two seemingly opposite statements: First, any information contained in the system's complete list of elements of reality the list is complete if it contains all possible predictions about the outcome of an experiment performed on the system is already contained in the system's wave function.
That is, the wave function includes all the elements of reality. The physicists formulated this statement in a paper last year. The second statement, which the physicists present here, is that a system's list of elements of reality includes its wave function. Taken together, the two statements imply that a system's wave function is in one-to-one correlation with its elements of reality.
By showing that the wave function fully describes reality, the argument also implies that quantum mechanics is a complete theory. If there was a one-to-one correspondence between the meteorologist's data and the weather, we would be in a very favorable situation: the forecast would then be as accurate as it can possibly be, in the sense that there does not exist any information that has not been accounted for.
In this sense, the wave function is an optimal description of reality. This argument is not the only one made recently in favor of the wave function's complete representation of reality. Pusey, Jonathan Barrett, and Terry Rudolph argued that the subjective interpretation of the wave function contradicts plausible assumptions in quantum mechanics, such as that multiple systems can be prepared in a way so that their elements of reality are uncorrelated.
While this approach is completely different from that of the current paper, the support from both papers may help point to an answer to one of the most long-standing debates in physics.
In the future, Colbeck and Renner plan to work on making the assumptions less stringent than they already are. However, it is certainly legitimate to question this 'free choice' assumption as well as the way 'free choice' is defined.
We are currently working on a proof that the assumption can be replaced by a weaker one which one might term 'partial freedom of choice'. Copyright Phys.
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