Schrodinger Wave Equation and Wave Function

The general one-dimensional Schrodinger Wave Equation is expressed as Where ψ(x,t) is the wave function, V(x) is the potential function and it is assumed to be independent of time. m is the mass of the particle and j is the imaginary number √(-1). The wave function ψ(x,t) is used to describe the behavior of the system mathematically ψ(x,t) can be a complex quantity.
The wave function ψ(x,t) can be rewritten as

 Where, ψ(x) is the function of position x and φ(t) is the function of time t.
Now the general form of Schrodinger Wave Equation can be rewritten as  Now left-hand side of the equation is only dependent upon the position x and right-hand side of the equation is dependent upon only time t. Each side of the equation must be equal to a constant quantity say η. This is because time-dependent side and position-dependent side of the equation is equal to each other.
Hence for the time-dependent side the equation would be,  Now, the solution is similar to classical exponential form of sinusoidal wave where η/(h/2π) = 2πη / h = ω = angular velocity of the sine wave.
Now as per quantum mechanics,  Separation constant of the equation is E, hence,  This is time independent form of Schrodinger Wave Equation.

Alternatively Establishing Time Independent Schrodinger Wave Equation

Now we will be trying to establish the time independent form Schrodinger Wave Equation and for that let us consider the wave equation  Now wavelength λ and momentum p of the wave are related to each other by the following equation called de Broglie wavelength equation,  Putting this value of λ in above second order differential equation we get,  Total energy of electron therefore,

Significance of Wave Function

We have already seen that the time and position dependent wave function ψ(x,t) can be rewritten as  As we have already proved, energy of electron E = η

In the year of 1926, Max Born stated that and postulated that if wave function of a particle is ψ(x,t), then probability of finding that particle between a gap of x and x + dx is,  Now we know some basic mathematics,  Therefore, |ψ(x,t)|2 can be written as,  Again as per basic complex mathematics,  Hence, it is proved that probability density function of a particle is independent of time. Hence, for finding position of electrons in crystal we should only concern with the time-independent wave function. It is needless to say that the probability of finding a particle anywhere in the universe is one. That means it must exist between the position - ∞ to + ∞. This convention then is mathematically represented in quantum mechanics or quantum physics by wave function as,

Closely Related Articles Op-amp | Working Principle of Op-ampAmplifier Gain | Decibel or dB GainIntegrated Circuits | Types of ICRegulated Power SupplyLaser | Types and Components of LaserWork FunctionMobility of Charge CarrierWhat are Photo Electrons? Electron volt or eVEnergy Quanta | Development of Quantum Physics Schottky EffectHeisenberg Uncertainty PrincipleCyclotron Basic Construction and Working PrincipleSinusoidal Wave SignalCommon Emitter AmplifierRC Coupled AmplifierDifferential AmplifierWave Particle Duality PrincipleSpace ChargeInverting AmplifierMore Related Articles Vacuum Diode History Working Principle and Types of Vacuum DiodePN Junction Diode and its CharacteristicsDiode | Working and Types of DiodeDiode CharacteristicsHalf Wave Diode RectifierFull Wave Diode RectifierDiode Bridge RectifierWhat is Zener Diode?Application of Zener DiodeLED or Light Emitting DiodePIN Photodiode | Avalanche PhotodiodeTunnel Diode and its ApplicationsGUNN DiodeVaractor DiodeLaser DiodeSchottky DiodePower DiodesDiode ResistanceDiode Current EquationIdeal DiodeReverse Recovery Time of DiodeDiode TestingMOSFET | Working Principle of p-channel n-channel MOSFETMOSFET CircuitsMOS Capacitor | MOS Capacitance C V CurveApplications of MOSFETMOSFET as a SwitchMOSFET CharacteristicsPower MOSFETHalf Wave RectifiersFull Wave RectifiersBridge RectifiersClamping CircuitTheory of SemiconductorIntrinsic SemiconductorExtrinsic SemiconductorsEnergy Bands of SiliconDonor and Acceptor Impurities in Semiconductor Conductivity of SemiconductorCurrent Density in Metal and Semiconductor Intrinsic Silicon and Extrinsic SiliconP Type SemiconductorN Type SemiconductorP N Junction Theory Behind P N JunctionForward and Reverse Bias of P N JunctionZener BreakdownAvalanche BreakdownHall Effect Applications of Hall EffectGallium Arsenide SemiconductorSilicon SemiconductorTypes of TransistorsBipolar Junction Transistor or BJTBiasing of Bipolar Junction Transistor or BJTTransistor BiasingTransistor CharacteristicsCurrent Components in a TransistorTransistor Manufacturing TechniquesApplications of Bipolar Junction Transistor or BJT | History of BJTTransistor as a SwitchTransistor as an AmplifierJFET or Junction Field Effect Transistorn-channel JFET and p-channel JFETApplications of Field Effect TransistorDIAC Construction Operation and Applications of DIACTRIAC Construction Operation and Applications of TRIACPhototransistorNew Articles Ring CounterDischarging a CapacitorCharging a CapacitorElectric PotentialParity GeneratorElectric Flux