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[nb 12][nb 13], As has been demonstrated, the set of all possible wave functions in some representation for a system constitute an in general infinite-dimensional Hilbert space. 20 In general, an even function times an even function produces an even function. well. But if the region in which the kinetic energy is ) \label{7.3}\], This probability is just the area under the function \(|Ψ(x,t)|^2\) between \(x\) and \(x+Δx\). compared to the energy of the particle). More, all α are in an n-dimensional set A = A1 × A2 × ... An where each Ai is the set of allowed values for αi; all ω are in an m-dimensional "volume" Ω ⊆ ℝm where Ω = Ω1 × Ω2 × ... Ωm and each Ωi ⊆ ℝ is the set of allowed values for ωi, a subset of the real numbers ℝ. They are, in a sense, a basis (but not a Hilbert space basis, nor a Hamel basis) in which wave functions of interest can be expressed. Due to the multiple possible choices of representation basis, these Hilbert spaces are not unique. particle can be written as the product some real function ψ(x) of space and This interpretation is possible because the square of the magnitude of a (a) Find the normalization constant. Beyond this interval, the amplitude of the wavefunction is zero because the ball is confined to the tube. It is single valued and continuous. In the corresponding relativistic treatment, In quantum field theory the underlying Hilbert space is, This page was last edited on 29 October 2020, at 07:02. L It is named after the function sine, of which it is the graph. In particular, the wavefunction is given by, \[\Psi \, (x,t) = A \, \cos \, (kx - \omega t) + i A \, \sin \, (kx - \omega t), \label{eq56}\], where \(A\) is the amplitude, \(k\) is the wave number, and \(ω\) is the angular frequency. The probability of finding the particle “somewhere” (the normalization condition) is, \[P(-\infty, +\infty) = \int_{-\infty}^{\infty} |\Psi \, (x,t)|^2 dx = 1.\label{7.4}\]. , 23 Integration of the square of the wavefunction over the last quarter of the tube yields the final answer. ( probability or transmittance is 0.001, i.e. The ball has a definite wavelength . An electron (m = 9.109*10-31 kg) is confined The complex conjugate of a function is obtaining by replacing every occurrence of in that function with . A clue to the physical meaning of the wave function is provided by the two-slit interference of monochromatic light ((Figure)). [The momentum operator in (Figure) is said to be the position-space representation of the momentum operator.] right shows. \end{align*} \], The average position of a large number of particles in this state is \(L/2\). (a) Find the normalization constant. someone moves it. Figure 3 displays the shape of the string at the times Third, if a matter wave is given by the wave function , where exactly is the particle? 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). If \(a = 3 + 4i\), what is the product \(a^*a\)? How is the wave function used to make predictions? \[(3 + 4i)(3 - 4i) = 9 - 16i^2 = 25 \nonumber\], Consider the motion of a free particle that moves along the x-direction. 0.05 t This means that the solutions to it, wave functions, can be added and multiplied by scalars to form a new solution. The function in the integral is a sine function with a wavelength equal to the width of the well, L—an odd function about \(x = L/2\). Bright fringes correspond to points of constructive interference of the light waves, and dark fringes correspond to points of destructive interference of the light waves (\(\PageIndex{1b}\)). (Part I), A ball is constrained to move along a line inside a tube of length \(L\). The square is used, rather than the modulus itself, just like the intensity of a light wave depends on the square of the electric field. (This is analogous to squaring the electric field strength—which may be positive or negative—to obtain a positive value of intensity.) k = √(p² + q²) tan a = sin a/cos a; In the Wave Equation, it is essential to have expert knowledge of. ∂ , Expectation-value calculations are often simplified by exploiting the symmetry of wave functions. The position operator introduces a multiplicative factor only, so the position operator need not be “sandwiched.”. (Later, we define the magnitude squared for the general case of a function with “imaginary parts.”) This probabilistic interpretation of the wavefunction is called the Born interpretation. where the x is sandwiched between the wave functions. (An odd function is also referred to as an anti-symmetric function.). Example \(\PageIndex{2A}\): Expectation Value (Part I), The normalized wavefunction of a particle is, \[\psi(x) = e^{-|x|/x_0} /\sqrt{x_0}. However, if the wavefunction does not vary slowly, we must integrate: \[P(x,x + \Delta x) = \int_x^{x + \Delta x} |\Psi \, (x,t)|^2 dx. Experimental measurements produce real (nonimaginary) numbers only, so the above procedure to use the wavefunction must be slightly modified. function, the Schroedinger equation does not have real, but The procedure for doing this is, where the quantity in parentheses, sandwiched between the wave functions, is called the momentum operator in the x-direction. Expectation Value (Part II) The time-dependent wave function of a particle confined to a region between 0 and L is. has touched your car. In this case, we have, Applying the normalization condition gives , so the wave function of the ball is. It is usually written, \[\langle x \rangle = \int_{-\infty}^{\infty} \Psi^* (x,t) \, x \Psi \, (x,t) \, dx. In general, the probability that a particle is found in the narrow interval \((x, x + dx)\) at time \(t\) is given by, \[P (x,x + dx) = |\Psi \, (x,t)|^2 dx = \Psi^* (x,t) \, \Psi \, (x,t) \, dx, \label{7.5}\]. Figure 5 displays the shape of the string at the times L The probability density of finding the system at time while the 3 black curves correspond to the states at times 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. A particle with mass m moving along the x-axis and its quantum state is represented by the following wave function: where . where is angular frequency and E is the energy of the particle. What precisely is “waving”? Notice that squaring the wave function ensures that the probability is positive. with energy hf = ∆E, or it can receive the required energy from another

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